Worked solution
Select indistinguishable beads by colour cases
A box contains 7 beads which are indistinguishable except for their colour, of which 3 are red,2 are blue,1 is white and 1 is green. Given that the order in which they are selected does not matter, how many ways can 3 beads be selected?
CJC · Lecture · Example 11
Case 1: three different colours
Choose 3 of the 4 colours: . Choosing a colour identifies the corresponding unlabelled bead type, not a labelled physical copy.
Case 2: two of one colour and one of a different colour
The repeated colour must be red or blue, giving choices. Then choose 1 of the other 3 colours: . The count is .
Case 3: all three have the same colour
Only red has at least 3 copies, so this contributes 1 selection: three red beads.
Add the disjoint multiplicity cases
Total number=. The colour multiplicity patterns 1+1+1,2+1, and 3 are mutually exclusive and exhaustive.
Common mistakes
Do not use 7 C3 as though the physical beads were all distinguishable. Count the colour selections subject to available multiplicities.