Evaluate \( { }_{11} P_{5} \) and \( { }_{11} \mathrm{C}_{4} \) \begin{tabular}{|}\( { }_{11} P_{5}= \) \\ \( { }_{11} C_{4}= \) \\ \end{tabular}

To evaluate \( { }_{11} P_{5} \) and \( { }_{11} C_{4} \), we can use the formulas for permutations and combinations. 1. The formula for permutations is: \[ { }_{n} P_{r} = \frac{n!}{(n-r)!} \] Thus, for \( { }_{11} P_{5} \): \[ { }_{11} P_{5} = \frac{11!}{(11-5)!} = \frac{11!}{6!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{1} = 55440 \] 2. The formula for combinations is: \[ { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \] For \( { }_{11} C_{4} \): \[ { }_{11} C_{4} = \frac{11!}{4!(11-4)!} = \frac{11!}{4! \times 7!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = \frac{7920}{24} = 330 \] Putting it all together, we get: \[ { }_{11} P_{5}= 55440 \\ { }_{11} C_{4}= 330 \]