جواب کاردرکلاس صفحه 128 درس 6 ریاضی دهم (شمارش، بدون شمردن)

ب  \(8! = 8\: \times \:\overbrace {7\: \times \:6\: \times \:5\: \times \:4\: \times \:3\: \times \:2\: \times \:1}^{7!} = 8\: \times \:7!\)

پ \(10! = 10\: \times \:\overbrace {9\: \times \:8\: \times \:7\: \times \:6\: \times \:5\: \times \:4\: \times \:3\: \times \:2\: \times \:1}^{9!} = 10\: \times \:9!\)

ت \(n! = n\: \times \overbrace {\:\left( {n - 1} \right)\: \times \:\left( {n - 2} \right)\: \cdots \: \times \:3\: \times \:2\: \times \:1}^{\left( {n - 1} \right)!} = n\: \times \:\left( {n - 1} \right)!\)

ب \(\frac{{10!}}{{9!}} = \frac{{10 \times 9!}}{{9!}} = 10\)

پ \(\frac{{n!}}{{\left( {n - 1} \right)!}} = \frac{{n \times \left( {n - 1} \right)!}}{{\left( {n - 1} \right)!}} = n\)

ت \(\frac{{8!}}{{6!}} = \frac{{8 \times 7 \times 6!}}{{6!}} = 8 \times 7 = 56\)

ث \(\frac{{10!}}{{8!}} = \frac{{10 \times 9 \times 8!}}{{8!}} = 90\)

ج \(\frac{{n!}}{{\left( {n - 2} \right)!}} = \frac{{n \times \left( {n - 1} \right)\left( {n - 2} \right)!}}{{\left( {n - 2} \right)!}} = \)

\(n\left( {n - 1} \right)\)

چ \(\frac{{8!}}{{5!}} = \frac{{8 \times 7 \times 6 \times 5!}}{{5!}} = \)

\(8 \times 7 \times 6 = 336\)

ح \(\frac{{10!}}{{7!}} = \frac{{10 \times 9 \times 8 \times 7!}}{{7!}} = 720\)

خ \(\frac{{n!}}{{\left( {n - 3} \right)!}} = \frac{{n \times \left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)!}}{{\left( {n - 3} \right)!}} = \)

\(n\left( {n - 1} \right)\left( {n - 2} \right)\)

د \(\frac{{n!}}{{\left( {n - 4} \right)!}} = \)

\(\begin{array}{l}\frac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right)!}}{{\left( {n - 4} \right)!}} = \\\\n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\end{array}\)

ذ \(\frac{{n!}}{{\left( {n - 5} \right)!}} = \)

ر \(\frac{{n!}}{{\left( {n - k} \right)!}} = \)

\(\begin{array}{l}\frac{{n\left( {n - 1} \right)\left( {n - 2} \right) \cdots \left( {n - k + 1} \right)\left( {n - k} \right)!}}{{\left( {n - k} \right)!}} = \\\\n\left( {n - 1} \right)\left( {n - 2} \right) \cdots \left( {n - k + 1} \right)\end{array}\)

ب \(9 \times 8 \times 7 \times 6 = \frac{{9\:!}}{{5\:!}}\)

پ \(11 \times 10 \times 9 = \frac{{11\:!}}{{8\:!}}\)

ت \(8 = \frac{{8\:!}}{{7\:!}}\)

ث \(n\left( {n - 1} \right) = \frac{{n\:!}}{{\left( {n - 2} \right)\:!}}\)

ج \(n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right) = \frac{{n\:!}}{{\left( {n - 4} \right)\:!}}\)