Evaluate the expression. a) \( { }_{7} C_{5} \) b) \( { }_{7} P_{5} \) a) \( { }_{7} C_{5}=\square \) (Simplify your answer.)

To evaluate the expression \( { }_{7} C_{5} \), we use the combination formula: \[ _{n} C_{r} = \frac{n!}{r!(n-r)!} \] For \( { }_{7} C_{5} \): \[ { }_{7} C_{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \] So, \( { }_{7} C_{5} = 21 \). Now, for \( { }_{7} P_{5} \) (permutation), we use the permutation formula: \[ _{n} P_{r} = \frac{n!}{(n-r)!} \] For \( { }_{7} P_{5} \): \[ { }_{7} P_{5} = \frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{7 \times 6 \times 5 \times 4 \times 3}{1} = 2520 \] Thus, \( { }_{7} P_{5} = 2520 \). Final answers: a) \( { }_{7} C_{5} = 21 \) b) \( { }_{7} P_{5} = 2520 \)