Choosing a PIN and a Committee
A school IT club is setting up a 4-digit PIN system for a locker. Digits may repeat, but the PIN cannot start with 0. Separately, the club must send a committee of 3 students to a competition from a group of 7 (4 boys and 3 girls). The advisor insists the committee must include at least one girl. A student claims that the number of valid PINs is 9 000 and that the number of valid committees is simply C(7,3) = 35, ignoring the gender rule. The advisor wants both claims checked and the correct counts justified.
1. Verify the claim that there are 9 000 valid 4-digit PINs (no leading zero, repetition allowed).
Read the worked answer
First digit: 9 choices (1–9). Each of the remaining three digits: 10 choices each. Total = 9 × 10 × 10 × 10 = 9 000. The claim is correct.
2. How many committees of 3 include at least one girl?
Read the worked answer
Total committees = C(7,3) = 35. Committees with no girl = choose all 3 from the 4 boys = C(4,3) = 4. At least one girl = 35 − 4 = 31. Alternatively: 1 girl + 2 boys = C(3,1)·C(4,2) = 3·6 = 18; 2 girls + 1 boy = C(3,2)·C(4,1) = 3·4 = 12; 3 girls = C(3,3) = 1. Sum = 18 + 12 + 1 = 31 ✓.
3. The student says the committee count is 35. Explain the error and give the correct number.
Read the worked answer
35 = C(7,3) counts all 3-person committees with no restriction, including the 4 all-boy committees. The gender rule forbids those 4, so the correct count is 35 − 4 = 31. The error was ignoring the 'at least one girl' condition.