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Calculate the value of 50C40

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  • Last Updated : 29 Oct, 2021

A permutation is termed as an arrangement of values in a definite order of a number of objects taken some or all at a time. For instance, let us assume to have first ten natural numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. If we need to compute the number of different unique 4-digit-PIN that can be formed using these 10 numbers. It is equivalent to 5040. P(10,4) = 5040.  The problem statement governing the calculation of permutations of 4 numbers taken from 10 numbers at a time is equal to the factorial of 10 divided by the factorial of the difference of 10 and 4. 

Therefore, we have, 

nPr = n!/(n-r)!

Combination

A combination problem statement is governed about grouping. Combinations help us calculate the number of different groups which can be formed from the available items. For instance, let us assume to have a team of 2 which is to be formed from 5 students(A, B, C, D, E). This is the combination of ‘r’ persons from the available ‘n’ persons is therefore given by the following equation: 

nCr = n!/r!(n-r)!

Uses of Permutation and Combination

A permutation finds its usage in the case of the list of data. The order of the data matters in case of computation of a permutation. A combination finds its usage in the case of a group of data. The order of data doesn’t matter in case of computation of a combination. 

Calculate the value of 50C40

Solution:

50C40 = n!/r!(n-r)!

= 50!/40!(50-40)!

= 50!/40!10!

= 50×49×48×47×46×45×44×43×42×41×40!/40!×10!

= 3.72E16/10!

50C40 = 10272278170

Sample Questions

Question 1. Difference between permutation and combination?

Solution:

PermutationCombination
Arranging people, digits, numbers, alphabets, letters, and colours Selection of menu, food, clothes, subjects, team.
Picking a team captain, pitcher and shortstop from a group. Picking three team members from a group.
Picking two favourite colours, in order, from a colour brochure. Picking two colours from a colour brochure.
Picking first, second and third place winners. Picking three winners.

Question 2. Compute the permutation and combination if n = 6 and r = 2

Solution:

We have the values, 

n = 6

r = 2

Computing the permutation, we have, 

nPr = n!/(n-r)!

= 6!/(6-2)!

= 6!/4!

= 6×5×4!/4!

6P2 = 30

For finding combination use the formula

nCr = n!/r!(n-r)!

= 6!/2!(6-2)!

= 6!/2!4!

= 6×5×4!/2!4!

= 6×5/2

6C2 = 15

Question 3. If Mallika has to choose 5 chocolates from 12. In how many ways she can choose?

Solution:

Here,

Mallika has to choose 5 chocolates from 12 thus the combination she can use is 12C5 ways.

12C5 = n!/r!(n-r)!

= 12!/5!(12-5)!

= 12!/5!7!

= 12×11×10×9×8×7!/5!7!

= 12×11×10×9×8/5!

= 792

Mallika can choose 5 chocolates out of 12 in 792 ways.

Question 4. Find the permutation of 10P4?

Solution:

nPr = n!/(n-r)!

10P4 = 10!/(10-4)!

= 10!/6!

= 10×9×8×7×6!/6!

10P4 = 5040

Question 5. Find the combination for 100C90?

Solution:

nCr = 100!/90!(100-90)!

= 100!/90!×10!

= 100×99×98×97×96×95×94×93×92×91×90!/90!×10!

100C90 = 17310309456440

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