The number of ways of choosing $n$ objects out of $3 n+1$ objects of which $n$ are identical and the

Question

The number of ways of choosing $n$ objects out of $3 n+1$ objects of which $n$ are identical and the rest are all different and different from the n objects is

Professor PC Thomas Classes · Answer

One possible selection are \ 1) $n$ identical in one way 2) $n-1$ from the identical ${ }^{2 n+1} C_1$ ways 3) $n-2$ from the identical ${ }^{2 n+1} C_2$ ways 0 from the identical ${ }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}}$ \ $ 	ext { Total }=1+{ }^{2 n+1} C_1+{ }^{2 n+1} C_2+\ldots+{ }^{2 n+1} C_n=2^{2 n} $