If n is an integer greater than 1 , then a- n C1(a-1)+ n C2(a-2)- ldots+(-1)n(a-n) is equal to :

Question

If n is an integer greater than 1 , then a- n C1(a-1)+ n C2(a-2)- ldots+(-1)n(a-n) is equal to : (A)  a  (B)  0  (C) a2 (D) 2n

Tardigrade Admin · Accepted Answer

a- n C1(a-1)+ n C2(a-2)- ldots+(-1)n(a-n) =a( n C0- n C1+ n C2- n C3+ ldots +(-1)n n Cn) +( n C1-2 n C2+3 n C3- ldots+(-1)n+1 n n Cn) ...(i) As we know (1-x)n= n C0-x n C1+x2 n C2-x3 n C3+ ldots+(-1)n xn n Cn ...(ii) Put x=1, we get 0= n C0- n C1+ n C2- n C3+ ldots+(-1)n n Cn On differentiating Eq. (ii) w.r.t. x, we get n(1-x)n-1=- n C1+2 x n C2-3 x2 n C3+ ldots +(-1)n n xn-1 n Cn Put x=1 0=- n C1+2n C2-3n C3+ ldots .+(-1)n-1 n n Cn From Eq. (i) a- n C1(a-1)+ n C2(a-2)- ldots+(-1)n(a-n) =a(0)+0=0

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

$a$

$0$

$a^{2}$

$2^{n}$

Q. If n is an integer greater than 1 , then a−nC1​(a−1)+nC2​(a−2)−…+(−1)n(a−n) is equal to :

a

0

a2

2n

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

$a$

$0$

$a^{2}$

$2^{n}$