# How many groups of four elements can be formed from nine elements .

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We need to find the number of groups that can be formed by taking 4 elements out of 9 that we have. Now while forming groups the order in which the elements are chosen does not matter, therefore we use the formula for combinations.

Now the expression for choosing n elements from r elements is C(n,r) = n!/ r!(n-r)!

Here n = 9 and r = 4.

So C(n,r) = C(9,4) = 9! / 4! * 5! =( 9*8*7*6)/ (4*3*2) = 126

**Therefore 126 groups can be formed.**

We'll write the formula of the combination of n elements taken k at a time:

**C(n,r) = n!/k!(n-k)!**

To determine the number of gropus of 4 that can be formed from 9 elements, we'll apply the combination formula:

C(9,4) = 9!/4!(9-4)!

C(9,4) = 9!/4!*5!

But 9! = 5!*6*7*8*9

4! = 1*2*3*4

C(9,4) = 5!*6*7*8*9/1*2*3*4*5!

We'll simplify and we'll get:

C(9,4) = 7*2*9/1

**C(9,4) = 126 groups of 4 that can be formed from 9 elements.**

There are 9 elements.

Let the number of ways we can group 4 elements from 9 elements be x.

The number of different ways can we arrange the 9 different elements in consecutive 4 places is 9P4 ways = 9*8*7*6 ways.

Alternatively , let number of ways of selection of 4 distinct elements from 9 elements be x . And each group could be arranged in 4! arrangements within itself. So x different groups of 4 elements could be arranged in 4!*x ways.

Therefore x*4! = 9*8*7*6.

Therefore x = 9*8*7*6/4!.

x = 9*8*7*6/4*3*2*1.

x = 126.

Therefore the number of ways selecting the group of 4 elements from 9 elements is 126.