Advertisements
Advertisements
प्रश्न
Write \[\sum^m_{r = 0} \ ^{n + r}{}{C}_r\] in the simplified form.
उत्तर
We know:
\[ \because \ ^{n}{}{C}_0 = \ ^{n + 1}{}{C}_0 \]
\[ \therefore \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 1}{}{C}_0 + \ ^{n + 1}{}{C}_1 + \ ^{n + 2}{}{C}_2 + \ ^{n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m \]
\[Using \ ^{n}{}{C}_{r - 1} + \ ^{n}{}{C}_r = \ ^{n + 1}{}{C}_r : \]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 2}{}{C}_1 + \ ^{n + 2}{}{C}_2 + \ ^{n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m \]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 3}{}{C}_2 + \ ^ {n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m\]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + m + 1}{}{C}_m\]
APPEARS IN
संबंधित प्रश्न
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?
A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?
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?
Twelve students complete in a race. In how many ways first three prizes be given?
How many three-digit numbers are there with no digit repeated?
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?
Evaluate the following:
n + 1Cn
Evaluate the following:
If nC4 = nC6, find 12Cn.
If 28C2r : 24C2r − 4 = 225 : 11, find r.
In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student?
In how many ways can a football team of 11 players be selected from 16 players? How many of these will
include 2 particular players?
How many different selections of 4 books can be made from 10 different books, if
there is no restriction;
A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?
Find the number of diagonals of , 1.a hexagon
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (ii) at least one boy and one girl?
Find the number of (i) diagonals
Determine the number of 5 cards combinations out of a deck of 52 cards if there is exactly one ace in each combination.
If 20Cr = 20Cr + 4 , then rC3 is equal to
There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is
Among 14 players, 5 are bowlers. In how many ways a team of 11 may be formed with at least 4 bowlers?
Find n if `""^6"P"_2 = "n" ""^6"C"_2`
There are 20 straight lines in a plane so that no two lines are parallel and no three lines are concurrent. Determine the number of points of intersection.
There are 8 doctors and 4 lawyers in a panel. Find the number of ways for selecting a team of 6 if at least one doctor must be in the team.
Four parallel lines intersect another set of five parallel lines. Find the number of distinct parallelograms that can be formed.
Find the value of 15C4
Find the value of 15C4 + 15C5
We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?
In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if they can be of any colour
In how many ways can a football team of 11 players be selected from 16 players? How many of them will include 2 particular players?
There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is 12C2 – 5C2.
There are 3 books on Mathematics, 4 on Physics and 5 on English. How many different collections can be made such that each collection consists of:
| C1 | C2 |
| (a) One book of each subject; | (i) 3968 |
| (b) At least one book of each subject: | (ii) 60 |
| (c) At least one book of English: | (iii) 3255 |
There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is to be formed. Find:
| C1 | C2 |
| (a) In how many ways committee: can be formed | (i) 10C2 × 19C3 |
| (b) In how many ways a particular: professor is included | (ii) 10C2 × 19C2 |
| (c) In how many ways a particular: lecturer is included | (iii) 9C1 × 20C3 |
| (d) In how many ways a particular: lecturer is excluded | (iv) 10C2 × 20C3 |
Number of selections of at least one letter from the letters of MATHEMATICS, is ______.
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then, the number of such arrangements is ______.
