Questions: Evaluate 13C5 and 13C8. 13C5=1287 (Type a whole number.) 13C8= (Type a whole number.)

Solution Steps

To solve the problem of evaluating the binomial coefficients \({ }_{13} \mathrm{C}_{5}\) and \({ }_{13} \mathrm{C}_{8}\), we can use the formula for combinations, which is given by:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Given that \({ }_{13} \mathrm{C}_{5} = 1287\), we need to evaluate \({ }_{13} \mathrm{C}_{8}\). Note that \(\binom{n}{k} = \binom{n}{n-k}\), so \({ }_{13} \mathrm{C}_{8} = { }_{13} \mathrm{C}_{5}\).

1. Use the property of combinations \(\binom{n}{k} = \binom{n}{n-k}\) to find \({ }_{13} \mathrm{C}_{8}\).
2. Since \({ }_{13} \mathrm{C}_{5} = 1287\), it follows that \({ }_{13} \mathrm{C}_{8} = 1287\).

We are given that \({ }_{13} \mathrm{C}_{5} = 1287\).

Using the property of combinations, we know that: \[ { }_{n} \mathrm{C}_{k} = { }_{n} \mathrm{C}_{n-k} \] Thus, we have: \[ { }_{13} \mathrm{C}_{8} = { }_{13} \mathrm{C}_{5} \]

Since \({ }_{13} \mathrm{C}_{5} = 1287\), it follows that: \[ { }_{13} \mathrm{C}_{8} = 1287 \]

Final Answer