Question

In how many ways can 14 identical toys be distributed among 3 boys such that everyone gets at least one toy and no two boys get an equal number of toys.
(a) 65
(b) 62
(c) 60
(d) 70

Answer 1

Hint:
 We have 14 identical toys and we have to distribute them among three boys. There are certain conditions given for distributing these toys. We will use these conditions to make a list of possible distributions of the toys among the three boys. Since the toys are identical, we can permute the possible distributions with respect to the number of boys and obtain the total number of ways of distribution.

Complete step by step answer:
We have 14 identical toys. We have to distribute these among 3 boys such that each boy gets at least one toy and no two boys get an equal number of toys.
Let us assume that we give one toy to one boy. Then, we can distribute the remaining toys among the other two boys in the following ways:
$\left\{ 1,2,11 \right\},\left\{ 1,3,10 \right\},\left\{ 1,4,9 \right\},\left\{ 1,5,8 \right\},\left\{ 1,6,7 \right\}$
Now, let us assume that we give 2 toys to one boy. Avoiding the same distributions listed above, we can distribute the remaining toys among the other two boys in the following ways:
$\left\{ 2,3,9 \right\},\left\{ 2,4,8 \right\},\left\{ 2,5,7 \right\}$
Next, we will assume that we give three toys to one boy and distribute the remaining toys among the other two boys. Avoiding repetitions from the above two lists, we can distribute the toys in the following ways:
$\left\{ 3,4,7 \right\},\left\{ 3,5,6 \right\}$
We cannot assume that we give four toys to one boy and make a list of possible distributions. Since the list will contain the same distributions from the above-mentioned ways. Therefore, we have a total of 10 ways of distributing the toys under given conditions. Now, there are three boys and we can permute the above-mentioned distributions among the three boys. The number of ways to do so is $3!=6$. Hence, the total number of ways for distributing the toys among three boys is $10\times 6=60$. Therefore, the correct option is C.

Note:

 It is important to understand the concept of counting the number of ways. It is always useful to try to count case by case and then add them together to obtain the total number of ways. Combinations, permutations are very useful while counting. We should also keep in mind the principle of counting while solving such type of questions.