Worked solution
Split repeated-vowel endpoints into disjoint cases
(a) Find the number of ways of arranging the word PRELIMINARY if the first and last letter is a vowel. marks
(b) A group of 10 students comprising 3 males and 7 females attended their graduation dinner. Find the number of ways of seating them at a round table with 10 chairs of different colours and no two male students seated next to each other. marks
[2011/RVHS/II/8]
CJC · Review problems · Question 7
Answer
(a) 1088640; (b) 1512000.
(a) One endpoint is I and the other A or E
PRELIMINARY has eleven letters, with R and I each repeated twice. Its vowels are A, E, I, I under the source’s usual vowel convention. Choose A–I or E–I (2 choices) and their endpoint order (2 choices). The middle nine letters have only the R pair repeated, giving .
(a) The endpoints are A and E
The middle nine letters contain both repeated pairs R,R and I,I. With 2 endpoint orders, .
(a) The endpoints are I and I
The endpoint I’s are identical and the middle repeats only R. Thus . The disjoint total is .
(b) Circular gaps and distinct-colour seats
Arrange the seven females circularly in ways. Choose three of their seven gaps and arrange the three males in ways, preventing any adjacent males. The ten differently coloured chairs distinguish rotations, so assign one reference student’s chair in 10 ways. Thus .
Common mistakes
Different-colour chairs act as labelled seats, even though the table is round.
Solution notes
All three printed vowel-endpoint cases, their multiplicity factors and the labelled-seat calculation retained. No marks invented.