Question 6 of 12

Question 6 (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The word ASSIGNMENT has 10 letters: A(1), S(2), I(1), G(1), N(2), M(1), E(1), T(1) — a multiset with two repeated letters (S and N, each appearing twice).

Find. The count of distinguishable arrangements under each of the five stated conditions.

Approach. Use the multiset-permutation formula $n!/(n_1!n_2!\cdots)$, fixing letters/blocks as required by each condition and permuting only what remains.

1. (a) Total arrangements. With 10 letters and two letters (S,N) repeated twice each: $$\frac{10!}{2!\,2!}=\frac{3{,}628{,}800}{4}=\boxed{907{,}200}$$
2. (b) Start with A, end with T. Fixing the first and last positions leaves the 8 middle letters S,S,I,G,N,M,E,N (still S×2, N×2) to permute freely: $$\frac{8!}{2!\,2!}=\frac{40{,}320}{4}=\boxed{10{,}080}$$
3. (c) Start with a vowel. The vowels A, I, E each occur exactly once, so there are 3 independent choices for the first letter; each choice leaves the remaining 9 letters (still containing S×2, N×2) to permute: $$3\times\frac{9!}{2!\,2!}=3\times\frac{362{,}880}{4}=3\times 90{,}720=\boxed{272{,}160}$$
4. (d) Block SIGN in this order. Glue S, I, G, N together as one fixed-order unit, consuming one of the two S's, the I, the G and one of the two N's. What remains is A, S, M, E, N, T (6 single letters, no repeats left) plus the SIGN block: 7 units total, all distinct, so they permute in $7!$ ways: $$\boxed{7!=5{,}040}$$
5. (e) The three vowels {A,I,E} as one block. Glue A, I, E into a single block, which can be internally ordered in $3!$ ways since all three are distinct. The remaining 7 letters are S,S,G,N,M,N,T (S×2, N×2) plus the vowel block: 8 units, permuted in $8!/(2!2!)$ ways, times the $3!$ internal orderings: $$\frac{8!}{2!\,2!}\times 3! = \frac{40{,}320}{4}\times 6 = 10{,}080\times 6=\boxed{60{,}480}$$