Let Pₙ denote the number of distinct ways in which n runners in a race can cross the finish line, where two or more runners may finish simultaneously (ties are allowed). Then which of the following values are correct? (Select all that apply.)

Question

Accepted Answer

P₄ = 75. Ordered Bell numbers count the number of ordered set partitions (weak orderings). Recurrence: Pₙ = sumₖ₌₁ⁿ C(n,k)*Pₙ₋ₖ. P₄ = C(4,1)*13 + C(4,2)*3 + C(4,3)*1 + C(4,4)*1 = 52+18+4+1 = 75. P₅ = C(5,1)*75+C(5,2)*13+C(5,3)*3+C(5,4)*1+C(5,5)*1 = 375+130+30+5+1 = 541.

Let Pₙ denote the number of distinct ways in which n runners in a race can cross the finish line, where two or more runners may finish simultaneously (ties are allowed). Then which of the following values are correct? (Select all that apply.)

Solution

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