Question

A person shops for 10 chocolates of 3 different types .How many selections of chocolates can he make

Answer 1

Answer:

66

Hello. I hope this helps you.  Have a great day!!

Step-by-step explanation:

Let the number of chocolates of each type be a, b and c.

So a + b + c = 10 and each of them is a number from 0, 1, 2, 3,....

It's actually easier to deal with positive numbers, so the option of 0 complicates things.  Fix this by setting

u = a+1,   v = b+1,   w = c+1.

Then u, v, w are numbers from 1, 2, 3, 4,....

and they satisfy  u+v+w = a+b+c+3 = 13.

# ways of making a chocolate selection

= # ways of choosing a, b, c from 0, 1, 2, 3,.... such that a + b + c = 10

= # ways of choosing u, v, w from 1, 2, 3,... such that u + v + w = 13.

This last one is what we count then.

Imagine thirteen 1s in a row, which means there are 12 spaces in between them.

 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1

We just need to choose two spaces to break this row into 3 sections: a value for u, a value for v and a value for w.

For instance, choosing the 4th and 7th spaces results in:

 1 _ 1 _ 1 _ 1      1 _ 1 _ 1      1 _ 1 _ 1 _ 1 _ 1 _ 1  --->   4 + 3 + 6 = 13

So

# ways of making a chocolate selction

= # ways of choosing u, v, w

= # ways of choosing 2 spaces from 12 spaces

= ( 12 × 11 ) / ( 2 × 1 )

= 6 × 11

= 66