Permutation and Combination Calculator

Permutations is the process of arranging object at a time. Follow the below provided manual step by step procedure to solve the permutations of grouped items.

• Take the total number of items and ordering those items
• The permutations formula is P(n, r) = n!/(n-r)!
• Where, n is the total number of elements
• r is the number of ordering elements.
• Substitute the n and r vales in the formula.
• Compute the math operation to the probability value.

Combination is a process of selecting n objects taken r at time.These are easy and simple steps to solve any kind of combination problems. Go through them and solve problems.

• Take n and r values from the given problem.
• Combination formula is C(n, r) = (n!)/(r!(n-r)!)
• Replace the values in the above formula and perform math operations to get the answer.

Examples

Question 1: From a group of 7 men and 6 women, five persons are to be selected with at least 3 men?

Solution:

The 3 chances of selecting five persons from 7 men and 6 women with at least 3 men are follows.

1. Selecting 5 men from 7 men = ⁷C₅

2. Selecting 4 men and 1 women = ⁷C₄ x ⁶C₁

3. Selecting 3 men and 2 women = ⁷C₃ x ⁶C₂

The total number of ways = ⁷C₅ + ⁷C₄ x ⁶C₁ + ⁷C₃ x ⁶C₂

We know that nCr = (n!)/(r!(n-r)!)

= 7! / ((7-5)! 5!) + 7! / (4! (7-4)!) x 6 + 7! / ((3!(7-3)!) x 6!/ ((2!(6-2)!)

= (7x6x5!) / (2! x 5!) + (7x6x5x4!) / (4! x 3!) x 6 + (7x6x5x4!) / (3!4!) x (6x5x4!) / (4!2!)

= (7x6) / (2x1) + (7x6x5) / (3x2x1) x 6 + (7x6x5) / (3x2x1) x (6x5) / (2x1)

= 21 + 210 + 525

= 756

Question 2: In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

Solution:

The word 'CORPORATION' has 11 letters.

Out of 11 letters it is having 5 vowels. They are 'O', 'O', 'A', 'I', 'O'.

These 5 vowels can be grouped and considered as a single letter i.e OOAIO.

The remaining number of letters are 7. In these seven letter 'R' occurs 2 times rest are different.

Number of ways of arranging these 7 letters is = 7! / 2!

= (7 x 6 x 5 x 4 x 3 x 2 x 1) / (2 x 1)

= 2520

The ways of arranging 5 vowels is 5! / 3!

= (5 x 4 x 3!) / 3!

= 20

Hence, the required answer is 2520 x 20 = 50400. Onlinecalculator.guru has a huge collection of online calculators for plenty of concepts in Maths, Physics, Chemistry. Explore it and make your calculations quite easy.

1. What is the relation between permutation and combination?

Permutation is the process of arranging objects or items in an order. Combinations are the different ways of selecting the ojects from a group or collect, here order of the objects does not matter.

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2. How do you calculate possible combinations?

To find the combinations, simply we use the formula nCr = n!/ (r!(n-r)!). Here n is the total number of items and r is the number of items choosen at a time. To find the probability of an event, we have to find the combinations.

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3. Consider a word ‘YOURSELEVES’. In how many ways the letter can be arranged if U and S always come together and ‘U’ always precedes ‘S’?

The word YOURSELEVES contains 11 letters out of which S occurs twice and E occurs thrice.

IF U and S are together, treat it as one letter. The remaining 10 letters can be arranged in 10! / (2!3!)

= (10x9x8x7x6x5x4x3!) / (2!3!)

= 302400 ways.

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4. What is the main difference between permutation and combination?

The main difference between permutations and combinations is ordering. With permutations we care about the elements offered, while with combinations we don't. For example, if your ATM pin number is 1234. If you enter another four digit number except 1234 that contains 1, 2, 3, 4 as numbers but not in the same order, it will say wrong pin.

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