5 A donut shop offers 4 type of donuts type abcd Find how ma

5. A donut shop offers 4 type of donuts: type a,b,c,d. Find how many different dozens in the following order are possible.

i. a>=1, b>=2, c>0, d>=0

ii. a=2,b=2

Solution

5.

(i)

In a dozen there are 12 donuts. In this case we have to keep a minimum of 1 donut, 2 donut, 1 donut and 0 donut of a , b, c and d type respectively.

To simplify the problem we will keep the minimum number of donuts required into respective places and then start again.

Thus now we require 8 (12-1-2-1-0=8) donuts and there are 4 types of donuts.

Now we will use the bar and star method to find the number of donuts. In this method we will consider the each bar to represent a donut. And the seperation will be represented by a star.

Thus one possible combination (3 type a, 2 type b , 1 type c and 2 type d ) can be represented as follows.

| | | * | | * | * | |

The number of variations in this above arrangement will give the number of different dozens possible.

Thus number of dozens possible = C(11,3) = 165

ii. a=2,b=2

In this case we have to fix the numbers of donuts for a and b. Thus we have 8 donuts left to be distributed into c and d. Number of ways n this case would be C(8,1) = 8