Advertisements
Advertisements
प्रश्न
Write \[\sum^m_{r = 0} \ ^{n + r}{}{C}_r\] in the simplified form.
Advertisements
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
Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?
Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
Prove that
A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?
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 three-digit numbers are there?
How many 9-digit numbers of different digits can be formed?
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?
Evaluate the following:
12C10
If 8Cr − 7C3 = 7C2, find r.
If 15Cr : 15Cr − 1 = 11 : 5, find r.
If n +2C8 : n − 2P4 = 57 : 16, find n.
If α = mC2, then find the value of αC2.
From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways can this be done?
How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?
A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?
Find the number of (i) diagonals
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: at least 3 girls?
How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?
If 20Cr + 1 = 20Cr − 1 , then r is equal to
If mC1 = nC2 , then
If\[\ ^{( a^2 - a)}{}{C}_2 = \ ^{( a^2 - a)}{}{C}_4\] , then a =
Three persons enter a railway compartment. If there are 5 seats vacant, in how many ways can they take these seats?
In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women?
If C0 + C1 + C2 + ... + Cn = 256, then 2nC2 is equal to
Find the number of ways of drawing 9 balls from a bag that has 6 red balls, 5 green balls, and 7 blue balls so that 3 balls of every colour are drawn.
How many committee of five persons with a chairperson can be selected from 12 persons.
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 no girls
A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ______.
Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on other side of the table. The number of ways in which the seating arrangements can be made is `(11!)/(5!6!) (9!)(9!)`.
The number of positive integers satisfying the inequality `""^(n+1)C_(n-2) - ""^(n+1)C_(n-1) ≤ 100` is ______.
A badminton club has 10 couples as members. They meet to organise a mixed double match. If each wife refers to p artner as well as oppose her husband in the match, then the number of different ways can the match off will be ______.
The no. of different ways, the letters of the word KUMARI can be placed in the 8 boxes of the given figure so that no row remains empty will be ______.
