diagram, as shown, or using permutations and the Fundamental Counting Principle.) 5. a) At least two girls means two, three, or four girls: b) committees c) If Jim and Nanci must be on the committee, there are nine people left for the remaining two positions: committees d) More boys than means three or four boys: committees 6. a) (n 2) (n 3) = 30 – – 5n + 6 = 30 n2 – 5n 24 = 0 n2 – – (n 8) (n + 3) = 0 – = 8 = n or n –3 = is extraneous n –3 = 8 n b) = 8, 4 n n ≥ = 60(nC2), so 4 and 2, so 4 nP4 n n n ≥ ≥ ≥ 7. If you place the T and K as required, this can happen only one way for each position. The remaining seven letters can then be arranged in between, keeping in mind that there are repeated letters: two A’s, two S’s, and two O’s:
diagram, as shown, or using permutations and the Fundamental Counting Principle.) 5. a) b) At least two girls means two, three, or four girls: committees c) If Jim and Nanci must be on the committee, there are nine people left for the remaining two positions: committees d) More boys than means three or four boys: committees 6. a) (n – 2) (n – 3) = 30 n2 – 5n + 6 = 30 n2 – 5n – 24 = 0 (n – 8) (n + 3) = 0 n = 8 or n = –3 n = –3 is extraneous n = 8 b) n = 8, n ≥ 4 nP4 = 60(nC2), so n ≥ 4 and n ≥ 2, so n ≥ 4 7. If you place the T and K as required, this can happen only one way for each position. The remaining seven letters can then be arranged in between, keeping in mind that there are repeated letters: two A’s, two S’s, and two O’s:
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