The mean of the values 0,1,2, ldots, n having corresponding weight nC0, nC1, nC2, ldots, nCn respectively, is

Question

The mean of the values 0,1,2, ldots, n having corresponding weight nC0, nC1, nC2, ldots, nCn respectively, is (A) (2n/(n+1)) (B) (2n+1/n(n+1)) (C) (n+1/2) (D) (n/2)

Tardigrade Admin · Accepted Answer

The required mean is barx=(0 ⋅ 1+1 ⋅ n C1+2 ⋅ n C2+3 ⋅ n C3+ ldots+n ⋅ n Cn/1+ n C1+ n C2+ ldots+ n Cn) =( displaystyle∑r=0n r ⋅ n Cr/ displaystyle∑r=0n n Cr)=( displaystyle∑r=1n r ⋅ (n/r) ⋅ n-1 Cr-1/ displaystyle∑r=0n n Cr) =(n displaystyle∑r=1n n-1 Cr-1/ displaystyle∑r-0n n Cr) =(n ⋅ 2n-1/2n)=(n/2)

Q. The mean of the values $0, 1, 2, \dots, n$ having corresponding weight $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ respectively, is

$\frac{2 ^{n}}{( n + 1 )}$

$\frac{2 ^{n + 1}}{n ( n + 1 )}$

$\frac{n + 1}{2}$

$\frac{n}{2}$

Q. The mean of the values 0,1,2,…,n having corresponding weight nC0​,nC1​,nC2​,…,nCn​ respectively, is

(n+1)2n​

n(n+1)2n+1​

2n+1​

2n​

The required mean is xˉ=1+nC1​+nC2​+…+nCn​0⋅1+1⋅nC1​+2⋅nC2​+3⋅nC3​+…+n⋅nCn​​ =r=0∑n​nCr​r=0∑n​r⋅nCr​​=r=0∑n​nCr​r=1∑n​r⋅rn​⋅n−1Cr−1​​ =r−0∑n​nCr​nr=1∑n​n−1Cr−1​​ =2nn⋅2n−1​=2n​

Q. The mean of the values $0, 1, 2, \dots, n$ having corresponding weight $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ respectively, is

$\frac{2 ^{n}}{( n + 1 )}$

$\frac{2 ^{n + 1}}{n ( n + 1 )}$

$\frac{n + 1}{2}$

$\frac{n}{2}$