A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2019 · Q711 Marks

The word TOADSTOOL has 9 letters.

(i)Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.[1]
(ii)Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.[4]
(iii)Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.[2]
(iv)Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] Evaluated 6! = 720
[B1] Accept unevaluated Total no of arrangements: 9! / (2!3!) = 30240
[B1] Accept unevaluated No with Ts together = 8! / 3! = 6720
[M1] correct or their identified total – their identified Ts together,m,n integers > 1
[A1] CAO 23520
[M1] their identified number of arrangements with T at ends / their identified total number of arrangements or (7!/3!) / (9!/(2!3!)), m,n integers > 1
[A1] Final answer 1/36 or 0.0278
[M1] ⁴Cx seen alone or ⁴Cx x k≥1, k an integer, 0<x<4
[A1] ⁴C₂ x k, k = 1 oe or ⁴C₁ x m, m 1 oe alone
[M1] Add 3 or 4 identified correct scenarios only, accept unsimplified
[A1] CAO, WWW (Total) = 15 Only dependent on 2nd M mark

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