34 Найти значение выражения: 1) \frac{A_{15}^9 - A_{15}^8}{A_{15}^7}; 2) \frac{A_{18}^{10} + A_{18}^{11}}{A_{18}^{9}}; 3) A_{12}^{4} - \frac{A_{12}^{5}}{A_{12}^{3}}; 4) \frac{A_{5}^{4} \cdot A_{10}^{4}}{A_{7}^{4}}.

1. $$\frac{A_{15}^9 - A_{15}^8}{A_{15}^7} = \frac{\frac{15!}{(15-9)!} - \frac{15!}{(15-8)!}}{\frac{15!}{(15-7)!}} = \frac{\frac{15!}{6!} - \frac{15!}{7!}}{\frac{15!}{8!}} = \frac{15!(\frac{1}{6!} - \frac{1}{7!})}{\frac{15!}{8!}} = \frac{\frac{1}{6!} - \frac{1}{7!}}{\frac{1}{8!}} = \frac{\frac{7}{7!} - \frac{1}{7!}}{\frac{1}{8!}} = \frac{\frac{6}{7!}}{\frac{1}{8!}} = \frac{6 \cdot 8!}{7!} = \frac{6 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} = 6 \cdot 8 = 48$$.
2. $$\frac{A_{18}^{10} + A_{18}^{11}}{A_{18}^{9}} = \frac{\frac{18!}{(18-10)!} + \frac{18!}{(18-11)!}}{\frac{18!}{(18-9)!}} = \frac{\frac{18!}{8!} + \frac{18!}{7!}}{\frac{18!}{9!}} = \frac{18!(\frac{1}{8!} + \frac{1}{7!})}{\frac{18!}{9!}} = \frac{\frac{1}{8!} + \frac{1}{7!}}{\frac{1}{9!}} = \frac{\frac{1}{8!} + \frac{8}{8!}}{\frac{1}{9!}} = \frac{\frac{9}{8!}}{\frac{1}{9!}} = \frac{9 \cdot 9!}{8!} = \frac{9 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} = 9 \cdot 9 = 81$$.
3. $$A_{12}^{4} - \frac{A_{12}^{5}}{A_{12}^{3}} = \frac{12!}{(12-4)!} - \frac{\frac{12!}{(12-5)!}}{\frac{12!}{(12-3)!}} = \frac{12!}{8!} - \frac{\frac{12!}{7!}}{\frac{12!}{9!}} = \frac{12!}{8!} - \frac{12! \cdot 9!}{7! \cdot 12!} = \frac{12!}{8!} - \frac{9!}{7!} = \frac{12!}{8!} - \frac{9!}{7!} = \frac{12!}{8!} - \frac{9 \cdot 7!}{7!} = \frac{12!}{8!} - 9 = \frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} - 9 = 9 \cdot 10 \cdot 11 \cdot 12 - 9 = 11880 - 9 = 11871$$.
4. $$\frac{A_{5}^{4} \cdot A_{10}^{4}}{A_{7}^{4}} = \frac{\frac{5!}{(5-4)!} \cdot \frac{10!}{(10-4)!}}{\frac{7!}{(7-4)!}} = \frac{\frac{5!}{1!} \cdot \frac{10!}{6!}}{\frac{7!}{3!}} = \frac{5! \cdot \frac{10!}{6!}}{\frac{7!}{3!}} = \frac{5! \cdot 10! \cdot 3!}{6! \cdot 7!} = \frac{5! \cdot 10! \cdot 3!}{6 \cdot 5! \cdot 7 \cdot 6!} = \frac{10! \cdot 3!}{6! \cdot 7 \cdot 6!} = \frac{10! \cdot 3!}{7!} = \frac{10! \cdot 6}{7!} = \frac{10!}{7!} \cdot 6 = \frac{10 \cdot 9 \cdot 8 \cdot 7!}{7!} \cdot 6 = 10 \cdot 9 \cdot 8 \cdot 6 = 4320$$.

Ответ: 1) 48; 2) 81; 3) 11871; 4) 4320