Question

Consider a random sequence x1,…,x7 of 7 numbers each chosen from the set {1,2,3,…,10} with replacement. What is the probability that max{x1,…,x7} occurs exactly once in x1,…,x7?

Answer 1

random sequence of 7 numbers can be chosen from set {1,2,3,....,10} with replacement in ways.

If max{x1,...,x7) = 10, then the number of ways 10 can appear only once in the random sequence is ways

if max{x1,...,x7) = 9, then the number of ways 9 can appear only once in the random sequence is ways

If max{x1,...,x7) = 8, then the number of ways 8 can appear only once in the random sequence is ways

if max{x1,...,x7) = 7, then the number of ways 7 can appear only once in the random sequence is ways

If max{x1,...,x7) = 6, then the number of ways 6 can appear only once in the random sequence is ways

if max{x1,...,x7) = 5, then the number of ways 5 can appear only once in the random sequence is ways

If max{x1,...,x7) = 4, then the number of ways 4 can appear only once in the random sequence is ways

if max{x1,...,x7) = 3, then the number of ways 3 can appear only once in the random sequence is ​​​​​​​ways

if max{x1,...,x7) = 2, then the number of ways 2 can appear only once in the random sequence is ​​​​​​​​​​​​​​ways

So the prob is