Solution — 2022 Question 8 Permutations & Combinations

(a)

\begin{matrix} Number of ways \\ = (\binom{16}{5}) \\ = 4368 ∎ \end{matrix}

(b)

\begin{matrix} Number of ways \\ = (\binom{10}{3}) \times (\binom{6}{2}) \\ = 1800 ∎ \end{matrix}

(c)

Since both Mr Wang and Mrs Wang must be included, we fix them in the team and choose the remaining 3 from the 12 people who are neither Mr Wang, Mrs Wang, nor their children.

\begin{matrix} Number of ways \\ = (\binom{12}{3}) \\ = 220 ∎ \end{matrix}

(d)

Case 1: One of Mr/Mrs Wang, 0 of their children

\begin{matrix} Number of ways \\ = (\binom{2}{1}) \times (\binom{12}{4}) \\ = 990 \end{matrix}

Case 2: One of Mr/Mrs Wang, 1 of their children

\begin{matrix} Number of ways \\ = (\binom{2}{1}) \times (\binom{2}{1}) \times (\binom{12}{3}) \\ = 880 \end{matrix}

Case 3: Both Mr/Mrs Wang, 0 of their children

\begin{matrix} Number of ways \\ = (\binom{2}{2}) \times (\binom{12}{3}) \\ = 220 \end{matrix}

Case 4: Both Mr/Mrs Wang, 1 of their children

\begin{matrix} Number of ways \\ = (\binom{2}{2}) \times (\binom{2}{1}) \times (\binom{12}{2}) \\ = 132 \end{matrix}

\begin{matrix} Required number of ways \\ = 990 + 880 + 220 + 132 \\ = 2222 ∎ \end{matrix}