How many different ways can 15 candy bars be distributed between Ram, Shyam, Ghanshyam and Balram, if Ram cannot have more than 5 candy bars and Shyam must have at least two? Assume all candy bars to be alike.

Hint:
Here in this question we will use the concept of combination to find the different ways of distributing candy bars. Firstly we will calculate all the possible ways of distributing candy bars among four of them. Then we will find the number of possible ways of distributing candy bars with at least 6 candy bars to Ram. Then we will find their difference will give us the number of ways to distribute the candy bars with at most 5 candy bars to Ram and at least 2 candy bars to Shyam.

Complete step by step solution:
We have to calculate the number of ways of distributing candy bars among four of them with the condition of giving Shyam at least 2 candy bars.
Number of remaining candy bars after giving 2 candy bars to Shyam \[ = 15 - 2 = 13\]
Now, number of possible ways of distributing the 13 candy bars in Ram, Shyam, Ghanshyam and Balram \[ = \dfrac{{(13 + 4 - 1)!}}{{13!3!}} = \dfrac{{16!}}{{13!3!}} = {}^{16}{{\rm{C}}_3} = 560\]…………..(1)
Now we have to calculate the number of ways of distributing candy bars among 4 of them with the condition of giving at least 6 candy bars to Ram and giving Shyam at least 2 candy bars.
Number of remaining candy bars after giving 6 to Ram and 2 candy bars to Shyam \[ = 15 - 6 - 2 = 7\]
Number of possible ways of distributing candy bars, at least 6 to Ram and at least 2 to shyam and remaining candy bars among the 4 of them \[ = \dfrac{{(7 + 4 - 1)!}}{{7!3!}} = \dfrac{{10!}}{{7!3!}} = {}^{10}{{\rm{C}}_3} = 120\]…………..(2)
Therefore, the difference between equation (1) and equation (2) will give us the number of possible ways of distributing candy bars with at most 5 candy bars to Ram and at least 2 candy bars to Shyam.
Number of possible ways of distributing candy bars with at most 5 candy bars to Ram and at least 2 candy bars to Shyam \[ = 560 - 120 = 440\]

Hence, there are 440 ways of distributing candy bars with giving Ram not more than 5 candy bars and giving Shyam at least 2 candy bars.

Note:
Here, we have used combinations to find the number of ways for distributing candies. Permutations may be defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are 123, 132, 213, 231, 321, 312.
So, Number of permutations of n things, taken r at a time, denoted by \[{}^{\rm{n}}{{\rm{P}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{(n}} - {\rm{r)}}!}}\]
Combinations may be defined as the various ways in which objects from a set may be selected. For example: The different selections possible from the numbers 1, 2, 3 taking 2 at a time, are 12, 23 and 31.
So, Number of combinations possible from n group of items, taken r at a time, denoted by \[^{\rm{n}}{{\rm{C}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{r}}!{\rm{(n}} - {\rm{r)}}!}}\]