# Division of Objects into Groups

## Division of Objects into Groups

Division of items into groups of unequal size

• Number of ways in which (m + n) distinct items can be divided into two unequal groups containing m and n items is $$\frac{\left( m+n \right)!}{m!n!}$$.
• The number of ways in which (m + n + p) items can be divided into unequal groups containing m, n, p items is $$^{m+n+p}{{C}_{m}}.\,{{\,}^{n+p}}{{C}_{n}}=\frac{\left( m+n+p \right)!}{m!n!p!}$$.
• The number of ways to distribute (m + n + p) items among 3 persons in the groups containing m, n and p items = Number of ways to divide x (Number of groups) $$=\frac{\left( m+n+p \right)!}{m!n!p!}\times 3!$$.

Division of objects into groups of equal size: The number of ways in which mn distinct objects can be divided equally into m groups each containing n objects and the order of the groups is not important is $$\left[ \frac{\left( mn \right)!}{{{\left( n! \right)}^{m}}} \right]\frac{1}{m!}$$.

* The number of ways in which mn different items can be divided equally into m group each containing n objects and the order of group is important is $$\left( \frac{\left( mn \right)!}{{{\left( n! \right)}^{m}}}\times \frac{1}{m!} \right)m!=\frac{\left( mn \right)!}{{{\left( n! \right)}^{m}}}$$.

Example:  In how many ways can a pack of 52 cards be divided equally among four players in order?

Solution: Here 52 cards are to be divided into four equal groups and the order of the groups is important so required number of ways = $$\left( \frac{52!}{{{\left( 13! \right)}^{4}}\times 4} \right)4!=\frac{52!}{{{\left( 13! \right)}^{4}}}$$.

Example: In how many can a pack of 52 cards be divided equally into four groups?

Solution:  Here order is not important the total number of ways is $$\frac{52!}{{{\left( 13! \right)}^{4}}4!}$$.

Division of identical objects into groups.

*  the total number of ways of dividing n identical items among r persons, each one of whom, can receive 0, 1, 2 or more items (≤n)  is n + r – 1cr₋₁.

*   the total number of ways of dividing n identical items among r  persons, each one of whom, receives at least one item is $$n-{{1}_{{{c}_{r-1}}}}$$.

*   the number 0 of ways in which n identical items can be divided into r groups so that no group contains less that m items and more than k (m < k) is coefficient of xn in the expansion of $${{\left( {{x}^{n}}+{{x}^{m+1}}+……{{x}^{k}} \right)}^{r}}$$.

Example:  in how many ways can 20 identical toys be distributed among 4 children I that each one gets at least 3 toys?

Solution:  Required number of ways

= coefficient of x²⁰ in $${{\left( {{x}^{3}}+{{x}^{4}}+….{{x}^{20}} \right)}^{4}}$$,

= coefficient of x⁸ in $${{\left( 1+x+…x{{+}^{17}} \right)}^{4}}$$,

= coefficient of x⁸ in $${{\left( \frac{1-{{x}^{18}}}{1-x} \right)}^{4}}$$,

= coefficient of x⁸ in (1 – x)⁻⁴

= 8 + 4 – 1c₄₋₁ [Coefficient of in (1 – x)⁻r = n + r – 1cr₋₁]

= 11c₃

= 165