If n is a positive integer greater than 1- thena- nC1a-1- nC2a-2-1na-n is equal to

We have, t_r+1=( 1)^r· ^nC_r·(a r) =( 1)^r(a·^nC_r r·^nC_r) =( 1)^r(a·^nC_r n·^n 1C_r 1) (∵ r· ^nC_r=n· ^n 1C_r 1) Putting r=0, 1, 2,⋯, n, we get t_1=a· ^nC_0 n ...

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Standard XII

Mathematics

Sum of Coefficients of All Terms

If n is a pos...

Question

If $n$ is a positive integer greater than $1$ , then
$a -^{n} C_{1} (a - 1) +^{n} C_{2} (a - 2) - \dots + (- 1)^{n} (a - n)$ is equal to

n

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a

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0

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None of these

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Solution

The correct option is C $0$
We have, $t_{r + 1} = (- 1)^{r} \cdot^{n} C_{r} \cdot (a - r)$
$= (- 1)^{r} (a \cdot^{n} C_{r} - r \cdot^{n} C_{r})$
$= (- 1)^{r} (a \cdot^{n} C_{r} - n \cdot^{n - 1} C_{r - 1})$
$(∵ r \cdot^{n} C_{r} = n \cdot^{n - 1} C_{r - 1})$
Putting $r = 0, 1, 2, \dots, n$ , we get
$t_{1} = a \cdot^{n} C_{0} - n \cdot 0$
$t_{2} = - (a \cdot^{n} C_{1} - n \cdot^{n - 1} C_{0})$
$t_{3} = a \cdot^{n} C_{2} - n \cdot^{n - 1} C_{1}$
$\dots \dots \dots$
$t_{n + 1} = (- 1)^{n} (a \cdot^{n} C_{n} - n \cdot^{n - 1} C_{n - 1})$

Adding,
$Sum = a [^{n} C_{0} -^{n} C_{1} +^{n} C_{2} -^{n} C_{3} + \dots + (- 1)^{n} \cdot^{n} C_{n}] + n [^{n - 1} C_{0} -^{n - 1} C_{1} +^{n - 1} C_{2} - \dots + (- 1)^{n} \cdot^{n - 1} C_{n - 1}]$
$= a \times 0 + n \times 0 = 0$

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If n is a positive integer greater than 1, then a− nC1(a−1)+ nC2(a−2)−⋯+(−1)n(a−n) is equal to

If $n$ is a positive integer greater than $1$ , then
$a -^{n} C_{1} (a - 1) +^{n} C_{2} (a - 2) - \dots + (- 1)^{n} (a - n)$ is equal to