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

To evaluate \( { }_{7} \mathrm{P}_{5} \), we use the formula for permutations, which is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!} \] Thus, \[ _{7}P_{5} = \frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 7 \times 6 \times 5 \times 4 \times 3 = 2520 \] Now, for \( { }_{9} \mathrm{C}_{6} \), we use the combination formula: \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \] Thus, \[ _{9}C_{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6! \cdot 3!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \] Putting it all together: \[ { }_{7} \mathrm{P}_{5} = 2520 \] \[ { }_{9} \mathrm{C}_{6} = 84 \]