Mixed counting problems

Harder problems are usually two or three easy steps in a row, or several separate cases. Decide what is chosen (C), what is arranged (P or !), and which cases to add.

Select, then arrange. How many 4-letter words can be made from the letters of EQUATION using 2 vowels and 2 consonants?

• EQUATION has 5 vowels (E, U, A, I, O) and 3 consonants (Q, T, N).
• Choose the letters: ⁵C₂ × ³C₂ = 10 × 3 = 30 ways.
• Arrange each choice of 4 letters: 4! = 24.
• Total 30 × 24 = 720.

At least / at most: list the cases. A committee of 4 is chosen from 5 women and 6 men, with at least 2 women.

• 2 women, 2 men: ⁵C₂ × ⁶C₂ = 10 × 15 = 150.
• 3 women, 1 man: ⁵C₃ × ⁶C₁ = 10 × 6 = 60.
• 4 women: ⁵C₄ = 5.
• Total 215. Check: all committees ¹¹C₄ = 330, and those with at most 1 woman number 15 + 100 = 115; 215 + 115 = 330.

No two together: use the gaps. Seat 4 boys and 3 girls in a row so that no two girls sit together.

• Seat the boys first: 4! = 24 ways. They leave 5 gaps (before, between and after them).
• Put the 3 girls into 3 of the 5 gaps, in order: ⁵P₃ = 60.
• Total 24 × 60 = 1440.

Dictionary order. Where does TEAM come among the arrangements of A, E, M, T in dictionary order?

• Words starting with A, with E and with M: 3! = 6 each, so 18 words come first.
• Among words starting with T: TAEM, TAME, then TEAM.
• So TEAM is word number 18 + 3 = 21.

All or none: when a condition says 'either both or neither', count each case separately and add, just like the 'at least' cases.