The weighted mean of first n natural numbers whose weights are equal to the number of selections out of n natural numbers of corresponding numbers, is
n⋅2n−12n−1
3n(n+1)2(2n+1)
(n+1)(2n+1)6
n(n+1)2
The required mean
x¯=1⋅nC1+2⋅nC2+3⋅nC3+…+n⋅nCn nC1+nC2+…+nCn=∑r=0n r⋅nCr∑r=1n nCr=∑r=1n r⋅nrn−1Cr−1∑r=1n nCr=n⋅2n−12n−1