At a birthday party, a mother prepares a plate of cookies for 8 children. There

Question

At a birthday party, a mother prepares a plate of cookies for 8 children. There are plenty of chocolate chip, peanut butter, and oatmeal cookies, but each child only gets one cookie.

a.) How many different plates can be prepared?

b.) How many different plates can be prepared if at least one of each kind of cookie is given out?

c.) How many different plates can be prepared if no one likes oatmeal cookies?

d.) How many different plates can be prepared if two children insist on getting peanut butter?

e.) How many different plates can be prepared if the dog got into the kitchen and there are only two chocolate chip cookies left?

Explanation / Answer

Dear... Given that, A mother prepares a plate of cookies for 8 children.    a.
Total number of plates can be prepared by combination of C( 10,8). b. In this ststement 3 of 8 objects are fixed. so choose the remaining 5 from among 3, with repetitions. If at least one of each kind of cookie is given out, then different plates can be prepared as C(7,5). c. From the given statement. So, choose 8 from 2 with repetitions. If no one likes oatmeal cookies, then different plates can be prepared by C( 9,8). d. From the given statement, 2 of 8 objects are fixed. So, choose the remaining 6 from among 3, with repetitions. If two children insist on getting peanut butter, then different plates can be prepared as C( 8,6). e.
If the dog got into the kitchen and there are only two chocolate chip cookies left then, different plates can be prepared by combinations of  C( 9,8) + C( 8,7) + C ( 7, 6) Here,  zero chocolate chip cookies used- choose 8 from 2 with repetitions + two chocolate chip cookies used- choose remaining 6 from among 2 with repetitions  

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