With a little thought (see Notes below) it can be seen that the number of ways of distributing n particles in three sublevels is w ( n , 3 ) = w ( n , 2 ) + w ( n − − 1 , 2 ) + ⋯ ⋯ + w ( 1 , 2 ) + w ( 0 , 2 ) so that w ( n , 3 ) = ∑ ∑ k = 0 n w ( n − − k , 2 ) = ∑ ∑ k = 0 n ( n − − k + 1 ) ! ( n − − k ) ! 1 ! = ( n + 2 ) ! n ! 2 ! where we have used the following theorem involving binomial coefficients : ∑ ∑ k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! .