Groups is a very important sub-topic in COMBINATORICS which has a lot of significance in study of Combinations. To know what basically a group is, let us take a simple example. Suppose we have 4 apples and we are asked to find the number of ways in which we can select two apples. We all know that it is **4C2** = 6. Let us denote each apple by A,B,C and D . So, all possible combinations are (A,B), (A,C), (A,D), (B,C), (B,D) and (C,D) .

When we chose (A,B) we were left with (C,D). When we chose (A,C) we were left with (B,D). When we chose (A,D) we were left with (B,C). So just by taking 3 combinations , we got all the 6 combinations. Let us put our any one selection of apples in box B1 and the remaining apples in box B2. Now, a group is a set which contains both B1 and B2. So, total number of combinations we can form is 6 but the total number of groups is just 3.

A single group can contain 'n' number of elements where 'n' is a natural number. In the above case , there were two elements , B1 and B2.

Now, given below are some basic formulas for groups which will help in solving many groups problems....

- The number of ways in which , (m+n) items be divided into two unequal groups containing 'm' and 'n' items is

\(\left( \begin{matrix} m+n \\ m \end{matrix} \right) =\left( \begin{matrix} m+n \\ n \end{matrix} \right) =\frac { (m+n)! }{ m!n! } \)

2.The number of ways in which 'mn' different items can be divided equally into 'm' groups each containing 'n' items in which order of groups is

a. Not Important

\(\frac { (mn)! }{ { n! }^{ m }m! } \)

b.Important

\(\frac { (mn)! }{ { n! }^{ m } }\)

3.The number of ways in which (m+n+p) things can be divided into 3 different groups containing (m),(n) and (p) things is

\(\frac { (m+n+p)! }{ m!n!p! } \)

4.The number of ways in which 'n' different things can be arranged into 'r' different groups is

\(n!\left( \begin{matrix} n-1 \\ r-1 \end{matrix} \right) \)

SORRY for any mistake in this note. Actually this is my first note And I insist to mention the mistakes in the comment section so that I can correct it. Please do Like if you think its good.

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