Evaluate \( { }_{6} \mathrm{C}_{1} \) and \( { }_{8} \mathrm{P}_{5} \) \( { }_{6} C_{1}=\square \) \( { }_{8} \mathrm{P}_{5}=\square \)

To evaluate \( { }_{6} \mathrm{C}_{1} \), we use the formula for combinations, which is given by \( { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 6 \) and \( r = 1 \): \[ { }_{6} C_{1} = \frac{6!}{1!(6-1)!} = \frac{6!}{1! \cdot 5!} = \frac{6 \cdot 5!}{1 \cdot 5!} = 6 \] For \( { }_{8} \mathrm{P}_{5} \), we use the formula for permutations, given by \( { }_{n} P_{r} = \frac{n!}{(n-r)!} \). Here, \( n = 8 \) and \( r = 5 \): \[ { }_{8} P_{5} = \frac{8!}{(8-5)!} = \frac{8!}{3!} = \frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3!}{3!} = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 = 6720 \] Thus, \[ { }_{6} C_{1} = 6 \] \[ { }_{8} P_{5} = 6720 \]