What is the probability of randomly arranging the letters in the word 'STATISTICS' so that all three S's are together?

Instruction: Treat all S's as a single unit and calculate the probability of this specific arrangement.

Context: This question tests the candidate's ability to handle permutations involving repeated elements and specific conditions.

Updated

Example Answer

The probability is 1/15. “STATISTICS” has ten letters: three S’s, three T’s, two I’s, one A, and one C. There are 10! divided by 3! times 3! times 2!, or 50,400, distinct arrangements.

To put the three S’s together, I treat SSS as one block. That leaves eight objects, still with three identical T’s and two identical I’s. Their distinct arrangements number 8! divided by 3! times 2!, or 3,360. Dividing favorable arrangements by all arrangements gives 3,360/50,400, which simplifies to 1/15. The other repeated letters matter in both counts, so they cancel in the probability.

Worked example

A second check: the three S positions form one of C(10, 3) = 120 equally likely sets. Eight sets are consecutive triples, so 8/120 = 1/15.

Why this works

The block method accounts for repeated letters consistently and produces a probability between zero and one.

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