How to determine the number of combinations?

Question

Let's say we have the following items:

$A_1$

$B_1, B_2, B_3$

$C_1, C_2, C_3, C_4$

$D_1, D_2$

$E_1$

$F_1, F_2$

How many combinations of four items can we make when there can only be zero or one items of each letter?

Answer 1

This looks like something we'll have to do a little bit "by hand".

What we'll do is enumerate various ways of getting four letters out of ABCDEF and then figure out how many ways we can get those four letters given varying numbers of each type of object. There are $\binom{6}{4} = 15$ ways to get four letters:

ABCD: 1*3*4*2 = 24 ways ABCE: 1*3*4*1 = 12 ABCF: 1*3*4*2 = 24 ABDE: 1*3*2*1 = 6 ABDF: 1*3*2*2 = 12 ABEF: 1*3*1*2 = 6 ACDE: 1*4*2*1 = 8 ACDF: 1*4*2*2 = 16 ACEF: 1*4*1*2 = 8 ADEF: 1*2*1*2 = 4 BCDE: 3*4*2*1 = 24 BCDF: 3*4*2*2 = 48 BCEF: 3*4*1*2 = 24 BDEF: 3*2*1*2 = 12 CDEF: 4*2*1*2 = 16

Add all these up and we get $244$ distinct ways to select four objects so that each is from a different category. This is out of $\binom{13}{4} = 715$ ways to select four objects without this restriction.