Seven boys and five girls are going to a county fair to ride the teacup ride. Each teacup seats four persons. Tickets are assigned to specific teacups on the ride. If 12 tickets for the numbered seats are given out at random, determine the probability that four boys are given the first four seats on the first teacup.

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Asked Mar 20, 2019
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Seven boys and five girls are going to a county fair to ride the teacup ride. Each teacup seats four persons. Tickets are assigned to specific teacups on the ride. If 12 tickets for the numbered seats are given out at random, determine the probability that four boys are given the first four seats on the first teacup.            

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Expert Answer

Step 1

Total number of possible arrangements:

There are 12 seats in the ride. The number of people going for the ride is: 7 (boys) + 5 (girls) = 12.

Each teacup has 4 seats. So, with a total of 12 seats, there are 3 teacups.

Imagine that the numbering of the seats is started from the first teacup and ends in the third teacup.

So, the seat numbers for the first teacup are 1, 2, 3 and 4; the seat numbers for the second teacup are 5, 6, 7 and 8; the seat numbers for the third teacup are 9, 10, 11 and 12.

When there are no restrictions, in case of a random assignment, seat number 1 can be assigned to any of the 12 people; seat number 2 can be assigned to any of the 11 remaining people; seat number 3 can be assigned to any of the 10 remaining people and so on.

Thus, the total number of ways in which the 12 persons can sit in the 12 seats without any restrictions is: 12!

Step 2

Number of arrangements for 4 boys in first teacup:

The restriction is that four boys are to be seated in the first teacup. There being 7 boys, in case of this restriction, the seat number 1 can be assigned to any of the 7 boys; seat number 2 can be assigned to any of the 6 remaining boys; seat number 3 can be assigned to any of the 5 remaining boys; seat number 4 can be assigned to any of the 5 remaining boys. Thus, the possible number of arrangements of boys in the f...

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Basic Probability

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