NCR Formula

Selecting things for an arrangement without keeping the sequential order of the arrangement. Its purpose is to provide possible configurations in which the order of the elements is irrelevant. 

Finding a factorial is required if one wishes to do the calculation of a combination. A factorial is a result of multiplying the given number by all of the positive integers that are either equal to or less than the provided number. An exclamation mark is a symbol that is most often used to symbolise a factorial. 

What is the Ncr Formula?

The NCR formula is used to determine the count of the many ways in which r things may be picked from n different items when the order is not considered. The representation of it as seen below.

nCr​=nPr​​ / r!= n!​ / r! (n-r)!

Derivation of Ncr Formula

If there are n items in total, then the number of permutations that may be formed using r of those things is given.

a] The creation of a combination of r components from a total of n elements using any of C (n, r) techniques.

b] Assemble and arrange these r components in any one of the r! possible ways.

The count of the different permutations that may be configured is determined using the multiplication principle.

P (n, r) = C (n, r) ⁢* r!

Applying the formula for permutations P (n, r) = n! / (n – r!) to the formula presented above, we get the following

n! / (n – r)! = C (n, r) r!

The count of combinations, denoted by C (n, r), is determined by using the formula C (n, r) = n! / [r! (n – r)!]

Solved example

• Find the number of ways to select 4 pens from 8 pens on the pen stand using the Ncr formula.

We have the option to select it in

8P4  = 1680 ways

There is no particular sequence necessary for selecting the pens since we may choose any of them at random from the several ways they might be chosen.

 Thus we divide 1680 by 4!. i.e. 1680/24 = 70 ways.

Hence, The number of ways to select 4 from 8 pens is 70.