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How does nPr and nCr work?

Last updated date: 24th Jul 2024
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Hint: Permutation and Combinations, the various ways in which objects from a set may be selected, generally without replacement to form subsets. The selection of subsets is called a permutation when the order of selection is a factor. A combination is not a factor.

Complete step-by-step solution:
nPr and nCr are the probability function that represents permutation and combinations. The formula of finding nPr and nCr is
$^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
$^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Here n is the total number of objects and r is the number of selected objects.
Generally nPr is used for permutation, representing selecting a group of ‘r’ objects from a group of ‘n’ number of objects. The order of objects matters in case of permutation. The formula of permutation is:
$^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
For example 7 books have to arrange 5 on the shelf. So we can solve it from permutation.
${ \Rightarrow ^7}{P_5} = \dfrac{{7!}}{{(7 - 5)!}}$
${ \Rightarrow ^7}{P_5} = \dfrac{{7!}}{{2!}}$
${ \Rightarrow ^7}{P_5} = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}$
${ \Rightarrow ^7}{P_5} = 2520ways$
And nCr is used for combinations representing selecting of objects from a group of objects where order of object does not matter.
$^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
For example 7 books have to arrange 5 on the shelf. So we can solve it by combination.
${ \Rightarrow ^7}{C_5} = \dfrac{{7!}}{{5!(7 - 5)!}}$
${ \Rightarrow ^7}{C_5} = \dfrac{{7!}}{{5! \times 2!}}$
${ \Rightarrow ^7}{C_5} = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{5 \times 4 \times 3 \times 2 \times 1 \times 2 \times 1}}$
${ \Rightarrow ^7}{C_5} = 21$

Note: Factorial is a function that multiplies a number by every number below it. The function is used, among other things, to find the number of ways ‘n’ objects can be arranged. Factorial may be indicated by ‘!’ by this sign. Factorial is used for non-negative real numbers. For a negative number it will be a complex number.