Question:
For $2 \le \, r \, \le \, n,\binom{n}{r}+2 \binom{n}{r-1}+\binom{n+2}{r}$ is equal to
For $2 \le \, r \, \le \, n,\binom{n}{r}+2 \binom{n}{r-1}+\binom{n+2}{r}$ is equal to
Updated On: June 02, 2025
- $\binom{n+1}{r-1}$
- $2 \binom{n+1}{r+1}$
- $2 \binom{n+1}$
- $2 \binom{n+1}{r}$
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The Correct Option is D
Solution and Explanation
$\binom{n}{r}+2 \binom{n}{r-1}+\binom{n}{r-2}=\bigg[\binom{n}{r}+\binom{n}{r-1}\bigg]$
+$\bigg[\binom{n}{n-r}+\binom{n}{n-r}\bigg]=\binom{n+1}{r} + \binom{n+1}{r-1}=\binom{n+1}{r}$
$[\because \, ^nC_r + \, ^nC_{r-1}= \, ^{n+1}C_r]$
+$\bigg[\binom{n}{n-r}+\binom{n}{n-r}\bigg]=\binom{n+1}{r} + \binom{n+1}{r-1}=\binom{n+1}{r}$
$[\because \, ^nC_r + \, ^nC_{r-1}= \, ^{n+1}C_r]$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.