The value of 15 C 02-15 C 12-15 C 22-15 C 152 iA- 15B- -15C- 0D- 51

The correct option is C 0 nbsp;Let f(n)=C_0^2 C_1^2+C_2^2 C_3^2+⋯+( 1)^n C_n^2 We know that f(n)= 0 , amp; if n is odd ( 1)^n/2 C_n/2 , amp; if n is eve ...

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Standard XII

Mathematics

Sum of Binomial Coefficients with Alternate Signs

The value of ...

Question

The value of $^{15} C_{0}^{2} -^{15} C_{1}^{2} +^{15} C_{2}^{2} - \dots -^{15} C_{15}^{2}$ is

15

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- 15

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0

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51

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Solution

The correct option is C $0$
Let $f (n) = C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + \dots + (- 1)^{n} C_{n}^{2}$
We know that
$f (n) = {\begin{matrix} 0, & if n is odd (- 1)^{n / 2} C_{n / 2}, & if n is even \end{matrix}$

Since, here $n$ is odd $(n = 15)$
$∴^{15} C_{0}^{2} -^{15} C_{1}^{2} +^{15} C_{2}^{2} - \dots -^{15} C_{15}^{2} = 0$

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Sum of Binomial Coefficients with Alternate Signs

Standard XII Mathematics

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The value of 15C20− 15C21+ 15C22−⋯− 15C215 is

The correct option is C 0 Let f(n)=C20−C21+C22−C23+⋯+(−1)n C2n We know that f(n)={0 ,if n is odd(−1)n/2 Cn/2 ,if n is even Since, here n is odd (n=15) ∴ 15C20− 15C21+ 15C22−⋯− 15C215=0

The value of $^{15} C_{0}^{2} -^{15} C_{1}^{2} +^{15} C_{2}^{2} - \dots -^{15} C_{15}^{2}$ is