Question

Use counting arguments from the preceding chapter.

My couch potato friend enjoys sitting in front of the TV and grabbing handfuls of 6 chocolates at random from his snack jar. Unbeknownst to him, I have replaced one of the 16 chocolates in his jar with a cashew. (He hates cashews with a passion.)

a) How many possible outcomes are there the first time he grabs 6 chocolates?


b). How many of these include the cashew?

Answer 1

Now there are 15 chocolates and 1 cashew

Question (a)

Number of possible outcomes are there the first time he grabs 6 chocolates

In his first grab, he can pick 1 cholcate from any of the 15 choclates. The number of ways of doing it = 15

In his second grab, he can pick 1 cholcate from any of the reamining 14 choclates since he has already grabbed 1 chocolate in his first grab The number of ways of doing it = 14

In his third grab, he can pick 1 cholcate from any of the reamining 13 choclates since he has already grabbed 2 chocolates in his first 2 grabs. The number of ways of doing it = 13

In his fourth grab, he can pick 1 cholcate from any of the reamining 12 choclates since he has already grabbed 3 chocolates in his first 3 grabs. The number of ways of doing it = 12

In his fifh grab, he can pick 1 cholcate from any of the reamining 11 choclates since he has already grabbed 4 chocolates in his first 4 grabs. The number of ways of doing it = 11

In his sixth grab, he can pick 1 cholcate from any of the reamining 10 choclates since he has already grabbed 5 chocolates in his first 5 grabs. The number of ways of doing it = 10

But these can be in any order, so i will make sure that i grab them irrespective of the order which i grab them by dividing the r! here we are grabiing 6 cholcoates. So we divide outcomes by 6*5*4*3*2*1

So the total number of ways in which he grabs 6 cholocates for the firs time = 15*14*13*12*11*10 / 6*5*4*3*2*1

= 3603600 / 720

= 5005

Number of possible outcomes in which the first time he grabs 6 chocolates = 5005

Question (b)

How many of these include the cashew

It is given that the cashew should be grabbed

Since there is only cashew, number of ways in which it can be grabbed = 1

Nowe we shpuld look at the number of outcomes of grabbing 5 chocolates from the remaining 15 cholocolates

Here in his frist grab, he has already picked 1 cashew, so we will start with second grab

In his second grab, he can pick 1 cholcate from any of the 15 choclates. The number of ways of doing it = 15

In his third grab, he can pick 1 cholcate from any of the reamining 14 choclates since he has already grabbed 1 chocolate in his second grab The number of ways of doing it = 14

In his fourth grab, he can pick 1 cholcate from any of the reamining 13 choclates since he has already grabbed 2 chocolates in his second and third grabs. The number of ways of doing it = 13

In his fifth grab, he can pick 1 cholcate from any of the reamining 12 choclates since he has already grabbed 3 chocolates in his second, third and fourth grabs. The number of ways of doing it = 12

In his Sixth grab, he can pick 1 cholcate from any of the reamining 11 choclates since he has already grabbed 4 chocolates in his second, third, fourth and fifth grabs. The number of ways of doing it = 11

But these can be in any order, so i will make sure that i grab them irrespective of the order which i grab them by dividing the r! here we are grabiing 5 cholcoates. So we divide outcomes by 5*4*3*2*1

So the total number of ways in which he grabs 1 cashew = 15*14*13*12*11 / 5*4*3*2*1

  = 3603600 / 720

  = 3003

Number of possible outcomes in which he grabs 1 cashew = 3003