Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 4 - Combinatorics and Mathematical Induction [Latest edition]

A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many ways the person can select either an Indian or a Chinese food?

There are 3 types of toy car and 2 types of toy train available in a shop. Find the number of ways a baby can buy a toy car and a toy train?

How many two-digit numbers can be formed using 1, 2, 3, 4, 5 without repetition of digits?

Three persons enter into a conference hall in which there are 10 seats. In how many ways they can take their seats?

In how many ways 5 persons can be seated in a row?

A mobile phone has a passcode of 6 distinct digits. What is the maximum number of attempts one makes to retrieve the passcode?

Given four flags of different colours, how many different signals can be generated if each signal requires the use of three flags, one below the other?

Four children are running a race:
In how many ways can the first two places be filled?

Four children are running a race:
In how many different ways could they finish the race?

Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if repetitions of digits is allowed

Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if repetitions of digits is not allowed

How many three-digit numbers are there with 3 in the unit place?
with repetition

How many three-digit numbers are there with 3 in the unit place? 
without repetition

How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5? if repetition of digits allowed

How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5? if the repetition of digits is not allowed

How many three-digit odd numbers can be formed by using the digits 0, 1, 2, 3, 4, 5? if the repetition of digits is not allowed

How many three-digit odd numbers can be formed by using the digits 0, 1, 2, 3, 4, 5? if the repetition of digits is allowed

Count the numbers between 999 and 10000 subject to the condition that there are no restriction

Count the numbers between 999 and 10000 subject to the condition that there are no digit is repeated

Count the numbers between 999 and 10000 subject to the condition that there are at least one of the digits is repeated

How many three-digit numbers, which are divisible by 5, can be formed using the digits 0, 1, 2, 3, 4, 5 if repetition of digits are not allowed?

How many three-digit numbers, which are divisible by 5, can be formed using the digits 0, 1, 2, 3, 4, 5 if repetition of digits are allowed?

To travel from a place A to place B, there are two different bus routes B₁, B₂, two different train routes T₁, T₂ and one air route A₁. From place B to place C there is one bus route say B₁, two different train routes say T₁, T₂ and one air route A₁. Find the number of routes of commuting from place A to place C via place B without using similar mode of transportation

How many numbers are there between 1 and 1000 (both inclusive) which are divisible neither by 2 nor by 5?

How many strings can be formed using the letters of the word LOTUS if the word either starts with L or ends with S?

How many strings can be formed using the letters of the word LOTUS if the word neither starts with L nor ends with S?

Count the total number of ways of answering 6 objective type questions, each question having 4 choices

In how many ways 10 pigeons can be placed in 3 different pigeon holes?

Find the number of ways of distributing 12 distinct prizes to 10 students?

Find the value of 6!

Find the value of 4! + 5!

Find the value of 3! – 2!

Find the value of 3! × 2!

Find the value of `(12!)/(9! xx 3!)`

Find the value of `(("n" + 3)!)/(("n" + 1)!)`

Evaluate `("n"!)/("r"!("n" - "r")!)` when n = 6, r = 2

Evaluate `("n"!)/("r"!("n" - "r")!)` when n = 10, r = 3

Evaluate `("n"!)/("r"!("n" - "r")!)` when for any n with r = 2

Find the value of n if (n + 1)! = 20(n − 1)!

Find the value of n if `1/(8!) + 1/(9!) = "n"/(10!)`

If `""^(("n"  – 1))"P"_3 : ""^"n""P"_4` = 1 : 10 find n

If `""^10"P"_("r" - 1)` = 2 × ⁶Pr, find r

Suppose 8 people enter an event in a swimming meet. In how many ways could the gold, silver and bronze prizes be awarded?

Three men have 4 coats, 5 waist coats and 6 caps. In how many ways can they wear them?

Determine the number of permutations of the letters of the word SIMPLE if all are taken at a time?

A test consists of 10 multiple choice questions. In how many ways can the test be answered if each question has four choices?

A test consists of 10 multiple choice questions. In how many ways can the test be answered if the first four questions have three choices and the remaining have five choices?

A test consists of 10 multiple choice questions. In how many ways can the test be answered if question number n has n + 1 choices?

A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.

What is the maximum number of different answers can the students give?

A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.

How will the answer change if each question may have more than one correct answers?

How many strings can be formed from the letters of the word ARTICLE, so that vowels occupy the even places?

8 women and 6 men are standing in a line. How many arrangements are possible if any individual can stand in any position?

8 women and 6 men are standing in a line. In how many arrangements will all 6 men be standing next to one another?

8 women and 6 men are standing in a line. In how many arrangements will no two men be standing next to one another?

Find the distinct permutations of the letters of the word MISSISSIPPI?

How many ways can the product a² b³ c⁴ be expressed without exponents?

In how many ways 4 mathematics books, 3 physics books, 2 chemistry books and 1 biology book can be arranged on a shelf so that all books of the same subjects are together

In how many ways can the letters of the word SUCCESS be arranged so that all Ss are together?

A coin is tossed 8 times, how many different sequences of heads and tails are possible?

A coin is tossed 8 times, how many different sequences containing six heads and two tails are possible?

How many strings are there using the letters of the word INTERMEDIATE, if the vowels and consonants are alternative

How many strings are there using the letters of the word INTERMEDIATE, if all the vowels are together

How many strings are there using the letters of the word INTERMEDIATE, if vowels are never together

How many strings are there using the letters of the word INTERMEDIATE, if no two vowels are together

Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many distinct 6-digit numbers are there?

Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many of these 6-digit numbers are even?

Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many of these 6-digit numbers are divisible by 4?

If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
GARDEN

If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
DANGER

Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the 85^th string?

If the letters of the word FUNNY are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, find the rank of the word FUNNY

Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4, and 5 repetitions not allowed?

Find the sum of all 4-digit numbers that can be formed using digits 0, 2, 5, 7, 8 without repetition?

If nC₁₂ = nC₉ find 21C_n

If `""^15"C"_(2"r" - 1) = ""^15"C"_(2"r" + 4)`, find r

If ⁿP_r = 720 and ⁿC_r = 120, find n, r

Prove that ¹⁵C₃ + 2 × ¹⁵C₄ + ¹⁵C₅ = ¹⁷C₅

Prove that `""^35"C"_5 + sum_("r" = 0)^4 ""^((39 - "r"))"C"_4` = ⁴⁰C₅

If `""^(("n" + 1))"C"_8 : ""^(("n" - 3))"P"_4` = 57 : 16, find the value of n

Prove that `""^(2"n")"C"_"n" = (2^"n" xx 1 xx 3 xx ... (2"n" - 1))/("n"!)`

Prove that if 1 ≤ r ≤ n then `"n" xx ""^(("n" - 1))"C"_("r" - 1) = ""^(("n" - "r" + 1))"C"_("r" - 1)`

A Kabaddi coach has 14 players ready to play. How many different teams of 7 players could the coach put on the court?

There are 15 persons in a party and if each 2 of them shakes hands with each other, how many handshakes happen in the party?

How many chords can be drawn through 20 points on a circle?

In a parking lot one hundred, one-year-old cars, are parked. Out of them five are to be chosen at random for to check its pollution devices. How many different set of five cars can be chosen?

How many ways can a team of 3 boys,2 girls and 1 transgender be selected from 5 boys, 4 girls and 2 transgenders?

Find the total number of subsets of a set with
[Hint: ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿC_n = 2ⁿ] 4 elements

Find the total number of subsets of a set with
[Hint: ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿCn = 2ⁿ] 5 elements

Find the total number of subsets of a set with
[Hint: ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿCn = 2ⁿ] n elements

A trust has 25 members. How many ways 3 officers can be selected?

A trust has 25 members. In how many ways can a President, Vice President and a Secretary be selected?

How many ways a committee of six persons from 10 persons can be chosen along with a chair person and a secretary?

How many different selections of 5 books can be made from 12 different books if, Two particular books are always selected?

How many different selections of 5 books can be made from 12 different books if, Two particular books are never selected?

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees a particular teacher is included?

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees a particular student is excluded?

In an examination a student has to answer 5 questions, out of 9 questions in which 2 are compulsory. In how many ways a student can answer the questions?

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of exactly 3 women?

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of at least 3 women?

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of at most 3 women?

7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also has 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’ s relatives?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?

Find the number of strings of 4 letters that can be formed with the letters of the word EXAMINATION?

How many triangles can be formed by joining 15 points on the plane, in which no line joining any three points?

How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line?

There are 11 points in a plane. No three of these lies in the same straight line except 4 points, which are collinear. Find, the number of straight lines that can be obtained from the pairs of these points?

There are 11 points in a plane. No three of these lie in the same straight line except 4 points which are collinear. Find the number of triangles that can be formed for which the points are their vertices?

A polygon has 90 diagonals. Find the number of its sides?

By the principle of mathematical induction, prove that, for n ≥ 1
1³ + 2³ + 3³ + ... + n³ = `(("n"("n" + 1))/2)^2`

By the principle of mathematical induction, prove that, for n ≥ 1
1² + 3² + 5² + ... + (2n − 1)² = `("n"(2"n" - 1)(2"n" + 1))/3`

Prove that the sum of the first n non-zero even numbers is n² + n

By the principle of Mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`

Using the Mathematical induction, show that for any natural number n ≥ 2,
`(1 - 1/2^2)(1 - 1/3^2)(1 - 1/4^2) ... (1 - 1/"n"^2) = ("n" + 1)/2`

Using the Mathematical induction, show that for any natural number n ≥ 2,
`1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 +2 + 3 + 4) + .... + 1/(1 + 2 +  3 + ... + "n") = ("n" - 1)/("n" + 1)`

Using the Mathematical induction, show that for any natural number n,
`1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) + ... + 1/("n"("n" + 1)*("n" + 2)) = ("n"("n" + 3))/(4("n" + 1)("n" + 2))`

Using the Mathematical induction, show that for any natural number n,
`1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`

Prove by Mathematical Induction that
1! + (2 × 2!) + (3 × 3!) + ... + (n × n!) = (n + 1)! − 1

Using the Mathematical induction, show that for any natural number n, x²ⁿ − y²ⁿ is divisible by x + y

By the principle of Mathematical induction, prove that, for n ≥ 1
`1^2 + 2^2 + 3^2 + ... + "n"^2 > "n"^2/3`

Use induction to prove that n³ − 7n + 3, is divisible by 3, for all natural numbers n

Use induction to prove that 5ⁿ⁺¹ + 4 × 6ⁿ when divided by 20 leaves a remainder 9, for all natural numbers n

Use induction to prove that 10ⁿ + 3 × 4ⁿ⁺² + 5, is divisible by 9, for all natural numbers n

Prove that using the Mathematical induction
`sin(alpha) + sin (alpha + pi/6) + sin(alpha + (2pi)/6) + ... + sin(alpha + (("n" - 1)pi)/6) = (sin(alpha + (("n" - 1)pi)/12) xx sin(("n"pi)/12))/(sin (pi/12)`

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The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

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In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct i

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The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

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The number of 5 digit numbers all digits of which are odd i

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In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways

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If `""^(("n" + 5))"P"_(("n" + 1)) = ((11("n" - 1))/2)^(("n" + 3))"P"_"n"`, then the value of n are

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The product of r consecutive positive integers is divisible b

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The number of five digit telephone numbers having at least one of their digits repeated i

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If `""^("a"^2 - "a")"C"_2 = ""^("a"^2 - "a")"C"_4` then the value of a is

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There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

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The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

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The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines

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Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ______

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Number of sides of a polygon having 44 diagonals is ______

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If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

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In a plane there are 10 points are there out of which 4 points are collinear, then the number of triangles formed is

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In ²ⁿC₃ : ⁿC₃ = 11 : 1 then

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`""^(("n" - 1))"C"_"r" + ""^(("n" - 1))"C"_(("r" - 1))` is

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The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

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The number of rectangles that a chessboard has ______

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The number of 10 digit number that can be written by using the digits 2 and 3 is

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If P_r stands for ^rP_r then the sum of the series 1 + P₁ + ²P₂ + ³P₃ + · · · + ⁿP_n is

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The product of first n odd natural numbers equals

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If ⁿC₄, ⁿC₅, ⁿC₆ are in AP the value of n can be

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1 + 3 + 5 + 7 + · · · + 17 is equal to