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Worked solution

Arrange PROBABILITY with adjacent or separated Is

If the letters of PROBABILITY are arranged at random, find the probability that the two Is are (i) together, (ii) separated.

CJC · Lecture · Self-Practice 6

(i) Treat II as one block

PROBABILITY has 11 letters, with two Bs and two Is. The total is 11!/(2!2!)11!/(2!2!). With the Is together, theIIblock and 9 other letters give 10 outer objects with two identical Bs, counted by 10!/2!. The ratio is (10!/2!)/(11!/(2!2!))=2/11(10!/2!)/(11!/(2!2!))=2/11.

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(ii) Use the complement of adjacency

With exactly two Is, separated means not adjacent, so the probability is 1−2/11=9/111-2/11=9/11. Both printed answers are retained. These derivations supplement the answer-only source; uniform distinct-arrangement weighting follows from the constant multiplicity of identical-letter copies.

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Common mistakes

Remember the repeated Bs as well as the repeated Is. Unlike three Es, two Is have only the alternatives adjacent or separated.

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