Evaluate \( { }_{7} \mathrm{P}_{5} \) and \( { }_{9} \mathrm{C}_{6} \). \( { }_{7} \mathrm{P}_{5}=\square \) \( { }_{9} \mathrm{C}_{6}=\square \)

Certainly! Let's evaluate each of the combinatorial expressions step by step. ### 1. Evaluating \( { }_{7} \mathrm{P}_{5} \) **Definition:** \[ { }_{n} \mathrm{P}_{k} = \text{Number of permutations of } n \text{ items taken } k \text{ at a time} \] \[ { }_{n} \mathrm{P}_{k} = \frac{n!}{(n - k)!} \] **Calculation:** \[ { }_{7} \mathrm{P}_{5} = \frac{7!}{(7 - 5)!} = \frac{7!}{2!} \] \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] \[ 2! = 2 \times 1 = 2 \] \[ { }_{7} \mathrm{P}_{5} = \frac{5040}{2} = 2520 \] **Result:** \[ { }_{7} \mathrm{P}_{5} = 2520 \] --- ### 2. Evaluating \( { }_{9} \mathrm{C}_{6} \) **Definition:** \[ { }_{n} \mathrm{C}_{k} = \text{Number of combinations of } n \text{ items taken } k \text{ at a time} \] \[ { }_{n} \mathrm{C}_{k} = \frac{n!}{k!(n - k)!} \] **Calculation:** \[ { }_{9} \mathrm{C}_{6} = \frac{9!}{6! \times (9 - 6)!} = \frac{9!}{6! \times 3!} \] \[ 9! = 362880 \] \[ 6! = 720 \] \[ 3! = 6 \] \[ { }_{9} \mathrm{C}_{6} = \frac{362880}{720 \times 6} = \frac{362880}{4320} = 84 \] **Result:** \[ { }_{9} \mathrm{C}_{6} = 84 \] --- ### Summary \[ { }_{7} \mathrm{P}_{5} = 2520 \] \[ { }_{9} \mathrm{C}_{6} = 84 \]