The number of ways of choosing n objects out of$(3n + 1)$objects of which n are identical and $(2n + 1)$are distinct, is

If we choose $k(0\le k\le n)$ identical objects, then we must choose$(n – k)$ distinct objects. 

This can be done in ${ }^{2n+1}{C}_{n-k}$ ways. Thus, the required number of ways

$=\sum _{k=0}^n{ }^{2n+1}{C}_{n-k}{=}^{2n+1}{C}_n{+}^{2n+1}{C}_{n-1}+\dots {+}^{2n+1}{C}_0=2^{2n}.$