順列 $_nP_r$ と階乗 $n!$ の値を計算する問題です。

1. 問題の内容

順列 $_nP_r$ と階乗 $n!$ の値を計算する問題です。

2. 解き方の手順

(1) $_8P_2 = \frac{8!}{(8-2)!} = \frac{8!}{6!} = 8 \times 7 = 56$

(2) $_6P_3 = \frac{6!}{(6-3)!} = \frac{6!}{3!} = 6 \times 5 \times 4 = 120$

(3) $_5P_1 = \frac{5!}{(5-1)!} = \frac{5!}{4!} = 5$

(4) $_{10}P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 10 \times 9 \times 8 \times 7 = 5040$

(5) $_{12}P_3 = \frac{12!}{(12-3)!} = \frac{12!}{9!} = 12 \times 11 \times 10 = 1320$

(6) $_4P_4 = \frac{4!}{(4-4)!} = \frac{4!}{0!} = 4 \times 3 \times 2 \times 1 = 24$

(7) $2! = 2 \times 1 = 2$

(8) $3! = 3 \times 2 \times 1 = 6$

(9) $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

(10) $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$

3. 最終的な答え

(1) 56

(2) 120

(3) 5

(4) 5040

(5) 1320

(6) 24

(7) 2

(8) 6

(9) 720

(10) 40320