166 Сравните числа: a) C и C; б) C и C; в)* C21 и C21; г)* C12 и C12.

a) \( C_5^2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \cdot 4}{2} = 10 \)

\( C_5^3 = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \cdot 4}{2} = 10 \)

Следовательно, \( C_5^2 = C_5^3 \)

б) \( C_7^2 = \frac{7!}{2!(7-2)!} = \frac{7!}{2!5!} = \frac{7 \cdot 6}{2} = 21 \)

\( C_7^5 = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = \frac{7 \cdot 6}{2} = 21 \)

Следовательно, \( C_7^2 = C_7^5 \)

в) \( C_{21}^1 = \frac{21!}{1!(21-1)!} = \frac{21!}{1!20!} = 21 \)

\( C_{21}^{20} = \frac{21!}{20!(21-20)!} = \frac{21!}{20!1!} = 21 \)

Следовательно, \( C_{21}^1 = C_{21}^{20} \)

г) \( C_{12}^5 = \frac{12!}{5!(12-5)!} = \frac{12!}{5!7!} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 792 \)

\( C_{12}^7 = \frac{12!}{7!(12-7)!} = \frac{12!}{7!5!} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 792 \)

Следовательно, \( C_{12}^5 = C_{12}^7 \)

Ответ: a) C = C; б) C = C; в) C = C; г) C = C

Ответ: a) C = C; б) C = C; в) C = C; г) C = C