**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 – 1_{cr₋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 x^{n} 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 – 1_{c₄₋₁} [Coefficient of in (1 – x)⁻^{r} = n + r – 1_{cr₋₁}]

= 11_{c₃}

= 165