RD Sharma solutions for Mathematics [English] Class 11 chapter 16 - Permutations [Latest edition]

Compute: 

(i)\[\frac{30!}{28!}\]

Compute:

Compute:

 L.C.M. (6!, 7!, 8!)

Prove that

Find x in each of the following:

Find x in each of the following:

Find x in each of the following:

Convert the following products into factorials:

5 · 6 · 7 · 8 · 9 · 10

Convert the following products into factorials: 

3 · 6 · 9 · 12 · 15 · 18

Convert the following products into factorials: 

(n + 1) (n + 2) (n + 3) ... (2n)

Convert the following products into factorials:

1 · 3 · 5 · 7 · 9 ... (2n − 1)

Which of the following are true:

(2 +3)! = 2! + 3!

Which of the following are true:

(2 × 3)! = 2! × 3!

Prove that: n! (n + 2) = n! + (n + 1)!

If (n + 2)! = 60 [(n − 1)!], find n. 

If (n + 1)! = 90 [(n − 1)!], find n.

If (n + 3)! = 56 [(n + 1)!], find n.

If \[\frac{(2n)!}{3! (2n - 3)!}\]  and \[\frac{n!}{2! (n - 2)!}\]  are in the ratio 44 : 3, find n.

Prove that: 

Prove that:

\[\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}\]

Prove that:

In a class there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class in a function. In how many ways can the teacher make this selection?

A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties and 5 pencil varieties, in how many ways can he select these articles?

From Goa to Bombay there are two roots; air, and sea. From Bombay to Delhi there are three routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there?

A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?

There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?

A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?

A letter lock consists of three rings each marked with 10 different letters. In how many ways it is possible to make an unsuccessful attempt to open the lock?

There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 2 each?

Given 7 flags of different colours, how many different signals can be generated if a signal requires the use of two flags, one below the other?

A team consists of 6 boys and 4 girls and other has 5 boys and 3 girls. How many single matches can be arranged between the two teams when a boy plays against a boy and a girl plays against a girl?

How many A.P.'s with 10 terms are there whose first term is in the set {1, 2, 3} and whose common difference is in the set {1, 2, 3, 4, 5}?

From among the 36 teachers in a college, one principal, one vice-principal and the teacher-incharge are to be appointed. In how many ways can this be done?

How many different five-digit number licence plates can be made if

first digit cannot be zero and the repetition of digits is not allowed,

How many different five-digit number licence plates can be made if

the first-digit cannot be zero, but the repetition of digits is allowed?

How many four-digit numbers can be formed with the digits 3, 5, 7, 8, 9 which are greater than 7000, if repetition of digits is not allowed?

Since the  number has to be greater than 8000, the thousand's place can be filled by only two digits, i.e. 8 and 9.
Now, the hundred's place can be filled with the remaining 4 digits as the repetition of the digits is not allowed.
The ten's place can be filled with the remaining 3 digits.
The unit's place can be filled with the remaining 2 digits.
Total numbers that can be formed = `2xx4xx3xx2=48`

How many odd numbers less than 1000 can be formed by using the digits 0, 3, 5, 7 when repetition of digits is not allowed?

How many different numbers of six digits each can be formed from the digits 4, 5, 6, 7, 8, 9 when repetition of digits is not allowed?

How many different numbers of six digits can be formed from the digits 3, 1, 7, 0, 9, 5 when repetition of digits is not allowed?

How many four digit different numbers, greater than 5000 can be formed with the digits 1, 2, 5, 9, 0 when repetition of digits is not allowed?

Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?

A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.

A customer forgets a four-digits code for an Automatic Teller Machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6 and 9. Find the largest possible number of trials necessary to obtain the correct code.

In how many ways can three jobs I, II and III be assigned to three persons A, B and C if one person is assigned only one job and all are capable of doing each job?

How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat?

If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes ?

A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?

How many three digit numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit may be repeated any number of times?

How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

How many 5-digit telephone numbers can be constructed using the digits 0 to 9. If each number starts with 67 and no digit appears more than once?

Three dice are rolled. Find the number of possible outcomes in which at least one die shows 5 ?

In how many ways can 4 prizes be distributed among 5 students, when
(i) no student gets more than one prize?
(ii) a student may get any number of prizes?
(iii) no student gets all the prizes?

Evaluate each of the following:

⁸P₃

Evaluate each of the following:

Evaluate each of the following:

⁶P₆ 

Evaluate each of the following:

P(6, 4)

If ⁿP₄ = 360, find the value of n.

Prove that:1 . P (1, 1) + 2 . P (2, 2) + 3 . P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1.

If ^n +5P_n +1 =\[\frac{11 (n - 1)}{2}\]^n +3P_n, find n.

Four letters E, K, S and V, one in each, were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?

How many words, with or without meaning, can be formed by using all the letters of the word 'DELHI', using each letter exactly once?

How many words, with or without meaning, can be formed by using the letters of the word 'TRIANGLE'?

There are two works each of 3 volumes and two works each of 2 volumes; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?

There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect, answers are there to this question?

How many 6-digit telephone numbers can be constructed with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number starts with 35 and no digit appears more than once?

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repeated? How many of these will be even?

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels come together?

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels never come together? 

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels occupy only the odd places?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the vowels are always together?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the letter G always occupies the first place?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the letters P and I respectively occupy first and last place?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the vowels always occupy even places?

How many permutations can be formed by the letters of the word, 'VOWELS', when

there is no restriction on letters?

How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with E?

How many permutations can be formed by the letters of the word, 'VOWELS', when

each word begins with O and ends with L?

How many permutations can be formed by the letters of the word, 'VOWELS', when

all vowels come together?

How many permutations can be formed by the letters of the word, 'VOWELS', when

all consonants come together?

How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?

m men and n women are to be seated in a row so that no two women sit together. if m > n then show that the number of ways in which they can be seated as\[\frac{m! (m + 1)!}{(m - n + 1) !}\]

How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if 4 letters are used at a time?

How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used at a time.

How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used but first is vowel.

Find the number of words formed by permuting all the letters of the following words:
INDEPENDENCE

Find the number of words formed by permuting all the letters of the following words:
INTERMEDIATE

Find the number of words formed by permuting all the letters of the following words:
ARRANGE

Find the number of words formed by permuting all the letters of the following words:

INDIA

Find the number of words formed by permuting all the letters of the following words:

PAKISTAN

Find the number of words formed by permuting all the letters of the following words:

RUSSIA

Find the number of words formed by permuting all the letters of the following words:
SERIES

Find the number of words formed by permuting all the letters of the following words:
EXERCISES

Find the number of words formed by permuting all the letters of the following words:
CONSTANTINOPLE

How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?

Find the total number of arrangements of the letters in the expression a³ b² c⁴ when written at full length.

How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?

How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's come together?

How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?

A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?

The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word 'SURITI'.

In how many ways can the letters of the word
"INTERMEDIATE" be arranged so that:the vowels always occupy even places?

In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:

the relative order of vowels and consonants do not alter?

The letters of the word 'ZENITH' are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word 'ZENITH'?

The number of permutations of n different things taking r at a time when 3 particular things are to be included is

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

The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is

How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ?

The number of different signals which can be given from 6 flags of different colours taking one or more at a time, is

The number of words from the letters of the word 'BHARAT' in which B and H will never come together, is

The number of six letter words that can be formed using the letters of the word "ASSIST" in which S's alternate with other letters is

The number of arrangements of the word "DELHI" in which E precedes I is

The number of ways in which the letters of the word 'CONSTANT' can be arranged without changing the relative positions of the vowels and consonants is

The number of ways to arrange the letters of the word CHEESE are

Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is

If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is

If in a group of n distinct objects, the number of arrangements of 4 objects is 12 times the number of arrangements of 2 objects, then the number of objects is

The number of ways in which 6 men can be arranged in a row so that three particular men are consecutive, is

A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is

The product of r consecutive positive integers is divisible by

If ^k + 5P_{k + 1} =\[\frac{11 (k - 1)}{2}\]. ^k + 3P_k , then the values of k are

The number of arrangements of the letters of the word BHARAT taking 3 at a time is

The number of words that can be made by re-arranging the letters of the word APURBA so that vowels and consonants are alternate is

The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is

The number of ways in which the letters of the word ARTICLE can be arranged so that even places are always occupied by consonants is

In a room there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amounts of illumination is