Solution — 2019 Question 6 Permutations & Combinations

(a)

Treat the 2 vowels (E, I) as a single block. There are then 8 units to arrange.

\begin{matrix} Number of arrangements \\ = 8! \times 2! \\ = 80640 ∎ \end{matrix}

(b)

Using the complement method: subtract the number of selections with no vowels from the total.

Total ways to choose 4 letters from the 9 letters of CHEMISTRY: $(\binom{9}{4}) = 126$

Ways with no vowels (choose 4 from the 7 consonants): $(\binom{7}{4}) = 35$

\begin{matrix} Ways with at least one vowel \\ = (\binom{9}{4}) - (\binom{7}{4}) \\ = 91 ∎ \end{matrix}