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 ≤ k ≤ n) identical objects, then we must choose (n – k) distinct objects. This can be done in  2n+1Cn−k ways. Thus, the required number of ways=∑k=0n 2n+1Cn−k=2n+1Cn+2n+1Cn−1+…+2n+1C0=22n