A probability problem by Pankaj Joshi

We can choose 0 or 1 or 2 or 3 or ...... or 6 candies from brand 1 , brand 2 , ..... or brand 6 .

So, let the number of candies we choose from brand 'i' be ${ b }_{ i }$ and the total number of the candies 6 . So the sum of ${ b }_{ 1 },{ b }_{ 2 },{ b }_{ 3 },{ b }_{ 4 },{ b }_{ 5 },{ b }_{ 6 },{ b }_{ 7 }\quad and\quad { b }_{ 8 }$ must be equal to 6.

Also, this can also happen that we choose no candies from any 1 brand or any 2 brands or any n brands where n<8 and n is a natural number.

Hence, ${ b }_{ 1 }+{ b }_{ 2 }+{ b }_{ 3 }+{ b }_{ 4 }+{ b }_{ 5 }+{ b }_{ 6 }+{ b }_{ 7 }+{ b }_{ 8 }=6$ where ${ b }_{ i }$ is a natural number.

The number of ways in which we can choose the candies is equivalent of finding total number of solutions to the above equation. So, the number of solutions for the above equation are ${ _{ n+r-1 }{ C }_{ r-1 } }$ where n=6 , r=8 . Thus, the number of ways in which we can choose 6 candies from 8 brands that are available are ${ _{ 13 }{ C }_{ 7 }=1716 }$

13C7 = 1716