Permutations & Combinations: INSPIRATION word arrang. — HCI 2025 H2 Math Prelim Paper 2

(a)(i) INSPIRATION has 11 letters with repeats: \(\text{I}\times 3\), \(\text{N}\times 2\). Number of arrangements \[ =\frac{11!}{2!\,3!}=3\,326\,400. \]

(a)(ii) Let \(A\): “the N's are together” and \(B\): “the I's are all separated”.

Number for \(A\) (N's together): Treat the two N's as a single block. Total objects \(=10\) (with three I's still repeating). \[ n(A)=\frac{10!}{3!}=604\,800. \]

Number for \(B\) (I's all separated): Arrange the other 8 letters first (including the two N's), then slot the I's into the gaps. \[ n(B)=\frac{8!}{2!}\times{}^{9}C_{3}=1\,693\,440. \]

Number for \(A\cap B\) (both N's together and I's separated): Arrange 7 letters (NN as a block + 6 others) and slot the 3 I's into the \(8\) resulting gaps. \[ n(A\cap B)=7!\times{}^{8}C_{3}=282\,240. \]

Number of ways the letters N must be together or the letters I must be separated, but not both \[\begin{aligned} &= 604\,800+1\,693\,440-2(282\,240) \\ &= 1\,733\,760. \end{aligned}\]

(b) The 11 cards consist of 3 I's, 2 N's, and 6 other distinct letters. The 8 cards left behind are all distinct if and only if the repeats are reduced to one of each, i.e. the 3 removed cards consist of 2 of the 3 I's and 1 of the 2 N's.

Required probability \[ = \frac{{}^{6}C_{6}\times {}^{3}C_{1}\times {}^{2}C_{1}}{{}^{11}C_{8}} \]

OR \[ = \frac{{}^{2}C_{1}\times {}^{3}C_{2}}{{}^{11}C_{3}} = \frac{2}{55}. \]

Method 2. \[ \frac{3}{11}\cdot\frac{2}{10}\cdot\frac{2}{9}\times\frac{3!}{2!}=\frac{2}{55}. \]