Question

In how many ways can 10 identical blankets be given to 3 beggars such that each receives at least one blanket?

Answer 1

Hint: Division and Distribution of Identical Objects
Case 1
Number of ways in which n identical things can be divided into r groups, if blank groups are allowed (here groups are numbered, i.e., distinct)
= Number of ways in which n identical things can be distributed among r persons, each one of them can receive 0,1,2 or more items
\[ = {\text{ }}{}^{\left( {n + r - 1} \right)}{C_{\left( {r - 1} \right)}}\]
Case 2
Number of ways in which n identical things can be divided into r groups, if blank groups are not allowed (here groups are numbered, i.e., distinct)
= Number of ways in which n identical things can be distributed among r persons, each one of them can receive 1,2 or more items
= \[{}^{\left( {n - 1} \right)}{C_{\left( {r - 1} \right)}}\]

Complete step-by-step answer:
We have 10 blankets and 3 beggars
Condition: - Each beggar much receives at least one blanket.
So, we will use Case 2
Here the identical objects are blankets so, n =10
And distinct groups are beggars so, r=3
Using formula for case 2
\[
  {}^{\left( {n - 1} \right)}{C_{\left( {r - 1} \right)}} = {}^{\left( {10 - 1} \right)}{C_{\left( {3 - 1} \right)}} \\
   \Rightarrow {}^9{C_2} = \dfrac{{9!}}{{\left( {9 - 2} \right)!2!}} \\
   \Rightarrow {}^9{C_2} = \dfrac{{9!}}{{\left( {9 - 2} \right)!2!}} \\
   \Rightarrow {}^9{C_2} = \dfrac{{9{\rm X}8}}{2} = 36 \\
 \]


So, there are 36 ways in which 10 identical blankets are given to 3 beggars such that each receives at least one blanket.

Note: Distribution of identical objects is quite wide and an important topic in permutation and combination. While distributing identical objects it does not matter which object is given to which person, what matters is how many objects are given to any person.