Statistics - Combination with replacement

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Each of several possible ways in which a set or number of things can be ordered or arranged is called permutation Combination with replacement in probability is selecting an object from an unordered list multiple times.

Combination with replacement is defined and given by the following probability function:

Formula

${^nC_r = \frac{(n+r-1)!}{r!(n-1)!} }$

Where −

  • ${n}$ = number of items which can be selected.

  • ${r}$ = number of items which are selected.

  • ${^nC_r}$ = Unordered list of items or combinations

Example

Problem Statement:

There are five kinds of frozen yogurt: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. What number of varieties will there be?

Solution:

Here n = 5 and r = 3. Substitute the values in formula,

${^nC_r = \frac{(n+r-1)!}{r!(n-1)!} \\[7pt] \ = \frac{(5+3+1)!}{3!(5-1)!} \\[7pt] \ = \frac{7!}{3!4!} \\[7pt] \ = \frac{5040}{6 \times 24} \\[7pt] \ = 35}$

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