A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2025 · Q610 Marks

Alissa has 10 different books from the series Squares and Circles. The books look similar except for their colour. There are 3 blue books, 2 red books, 2 yellow books, 1 orange book, 1 purple book and 1 green book. Alissa places the books in a row on her shelf. She is only interested in the arrangement of the colours.

(a)How many different colour arrangements are there of the 10 books?[1]
(b)How many different colour arrangements are there of the 10 books in which the 3 blue books are together, but the 2 yellow books are not next to each other?[2]
(c)How many different colour arrangements are there of the 10 books with exactly 4 books between the 2 yellow books?[3]
(d)Alissa selects 4 books from her 10 different books from the series Squares and Circles. Find the number of different selections if the 4 books include at least 1 red book, at most 1 blue book and exactly 1 yellow book.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

[1] 10! / (3!2!2!) = 151200
CAO.
[1] 8!/(2!2!) - 7!/(2!) = 10080 - 2520 = 7560
[1] 7560
[1] Identify 5 possible pairs of positions for the yellow books, e.g., (1st and 6th, 2nd and 7th, ..., 5th and 10th).
[1] Calculate arrangements for the remaining 8 books: 8!/(3!2!) = 3360.
[1] Multiply the number of positions by the arrangements of other books: 5 × 3360 = 16800.
[1] One correct identified unsimplified expression (e.g. 2C1 × 2C1 × 3C1 × 3C1 = 36 or 2C1 × 2C2 × 3C1 × 3C0 = 6 or 2C1 × 2C1 × 3C0 × 3C2 = 12 or 2C1 × 2C2 × 3C0 × 3C1 = 6).
[1] Correct outcome/value for 2 clearly identified scenarios, accept unsimplified WWW.
[1] Sum of 4 correct identified scenarios (1Y1R1B1O, 1Y2R1B, 1Y1R2O, 1Y2R1O).
[1] 60

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