Question

There are four youth in my son’s Cub Scout den: Russ, Nicky, Josh, and Colleen.

1. Each child will build and enter a car in the pack’s annual Pinewood Derby race. There are two types of pinewood derby kits: cars and trucks. Each of the kids was asked whether they wanted to build a car or a truck. In how many different ways could they have answered? For example, one possibility is that Russ and Nicky want to build cars, while Josh and Colleen want to build trucks. Another possibility is that all four children want to build cars. And so on. Try to list and count the number of possibilities.

2.) Marion, the den leader, goes to the Scout shop and buys four kits: a race car, a police car, a fire truck, and a pickup truck. In how many different ways can the kits be distributed among the youth?

For example, one possibility is that Russ gets the police car, Nicky gets the race car, Josh gets the pickup truck, and Colleen gets the fire truck. Another possibility is that Russ gets the race car, Nicky gets the fire truck, Josh gets the pickup truck, and Colleen gets the police car. And so on. Note that Mr. Marion bought only one of each kit, so if Nicky gets the race car, then none of the other three can have a race car as well. Try to list and count the number of possibilities.

3.) The day of the race is here! All of the kits are built and ready to race. In the den, ribbons will be given for first, second, and third place; a blue ribbon for first place, a red ribbon for second place, and a white ribbon for third place. In how many different ways can the kids win the ribbons? For example, one possibility is that Russ gets the blue ribbon, Nicky gets the red ribbon, and Colleen gets the white ribbon. Another possibility is that Nicky gets the blue ribbon, Colleen gets the red ribbon, and Josh gets the white ribbon. And so on. Try to list and count all of the possibilities.

4.) In addition to winning ribbons, the top two finishers in each den are entered into the pack championship, to race for the big trophy. In how many different ways can a pair of kids be chosen to represent this den in the pack championship? For example, one possibility is that Russ and Colleen advance to the championships. Another possibility is that Nicky and Josh advance. And so on. Note that “Russ and Colleen” counts as the same pair as “Colleen and Russ”; the order in which they finished in the den race doesn’t matter. Try to list and count all of the possible pairs.

Answer 1

There are four youths here. They are Russ (R), Nicky (N), Josh (J), and Colleen (C).

1. There are two kits here: one for trucks and one for cars. The choice of each individual can be either building a truck or car. Also, the choice of each kid will be independent of the other. Thus, we can list the combinations as:

Build Car	Build Truck
R, N, J, C	No one
R, N, J	C
R, N, C	J
R, J, C	N
N, J, C	R
R, N	J, C
R, J	N, C
R, C	N, J
N, J	R, C
N, C	R, J
J, C	R, N
No one	R, N, J, C
C	R, N, J
J	R, N, C
N	R, J, C
R	N, J, C

By counting, we can see that this is a list of a total of combinations.

2. There are four kits purchased: a race car, a police car, a fire truck, and a pickup truck. Each of the individuals shall get one car. This can happen in the following ways:

Race Car	Police Car	Fire Truck	Pickup Truck
R	N	J	C
R	N	C	J
R	J	N	C
R	J	C	N
R	C	N	J
R	C	J	N
N	R	J	C
N	R	C	J
N	J	R	C
N	J	C	R
N	C	J	R
N	C	R	J
C	N	R	J
C	N	J	R
C	J	N	R
C	J	R	N
C	R	J	N
C	R	N	J
J	N	R	C
J	N	C	R
J	R	N	C
J	R	C	N
J	C	R	N
J	C	N	R

By counting, we see that the total number of possibilities is 24.

3. There are three ribbons to be given namely, blue for the first position, red for the second position and white for the third position. There are a total of four candidates for the ribbons. Each candidate can be given a maximum of one ribbon and each ribbon can be given to one candidate. Seeing all combinations, we get the following list:

Blue	Red	White
R	N	J
R	N	C
R	J	N
R	J	C
R	C	N
R	C	J
N	R	J
N	R	C
N	J	R
N	J	C
N	C	J
N	C	R
C	N	R
C	N	J
C	J	N
C	J	R
C	R	J
C	R	N
J	N	R
J	N	C
J	R	N
J	R	C
J	C	R
J	C	A

As can be seen, there a total of 24 combinations.

4. The top two candidates can qualify for the championship. Here, the order does not matter and hence J and R is the same as R and J. Therefore, we get the following list of the qualifiers:

Qualifiers

R and N

R and J

R and C

N and J

N and C

J and C

Thus, there a total of 6 combinations.