A-LevelMathematicsProbability and statistics9709/52 · Oct/Nov 2025 · Q710 Marks

The 10 letters in the word ZOOLOGICAL are Z, O, O, L, O, G, I, C, A, L.

(a)Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which the three Os are together and the two Ls are not next to each other.[4]
(b)Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which there are exactly 5 letters between the two Ls.[3]
(c)Two letters are chosen at random from the 10 letters in the word ZOOLOGICAL. Find the probability that these two letters are different.[3]

Mark scheme

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

[B1] 8!/2! seen alone (not multiplied/divided).
[B1] b-7!, 5040 < b.
[M1] 8!/(c!d!) - 7!/(c!d!) c=1,2 d=1,3.
[A1] CAO.
[B1] 6!xe, 1<e<=42.
[B1] fx(7x6)/2, 1<f accept 7C2 or 7P2/2.
[M1] 6!/g! * 7x6/h! g=1,3 h=1,2,3.
[A1] CAO.
[B1] 8!/3!xi, i>=1.
[M1] 8!/j! * 4, j=1,2,3.
[A1] CAO.
[B1] 8Pm x k, k>=1.
[M1] 8Pm x 4!/3! or 8P5 x 4, m=3,4,5. or 8C5 x 4!/3! or 8C5 x 4.
[A1] CAO.
[M1] 3C2 + 2C2 seen.
[M1] 1 - f/10C2, 1<=f<45.
[A1] If one or more M mark not awarded, SCB1 for 41/45, 0.911 WWW.
[M1] 3/10*2/9 + 2/10*1/9 seen.
[M1] 1 - (3/10*2/9 + 2/10*1/9) d=9 or 10. Accept 1-(6/90 + 2/90) or 1-(3/45+1/45) or 1-(2/30+1/45).
[A1] If one or more M mark not awarded, SCB1 for 41/45, 0.911 WWW.
[M1] 3C1 x 2C1 + 3C1 x 5C1 + 5C1 x 2C1 + 5C2 seen
[M1] g/10C2, 1<=g<45
[A1] If one or more M mark not awarded, SCB1 for 41/45, 0.911 WWW.

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