A large university is planning to build a physical education complex and wants t

Question

A large university is planning to build a physical education complex and wants to know how to divide space for racquetball, squash, and tennis courts. To that end, 250 students were selected randomly and it was found that among these: 55 play racquetball, 25 play squash, 65 play tennis, 15 play racquetball and squash, 10 play tennis and squash, 25 play tennis and racquetball, 5 play all three games.

a) How many of the students surveyed play none of the three games?

b) How many play racquetball and tennis but not squash?

Explanation / Answer

Let Racquetball be represented as R , Squash be represented as S and Tennis be represented as T.

Then we are given that :

n(R) = 55, n(S) = 25, n(T) = 65, n( R and S) = 15, n( T and S ) = 10, n( R and T ) = 25, n( R and T and S) = 5

a) Now as we know that 5 play all three games, therefore, from this we get

n ( R and S and no T ) = n(A and S ) - n( R and T and S) = 15 - 5 = 10

Similarly,

n ( T and S and no R ) = n(T and S ) - n( R and T and S) = 10 - 5 = 5

n ( R and T and no S ) = n(R and T ) - n( R and T and S) = 25 - 5 = 20

. Using the same concept we get:

n ( only R ) = n(R) - n ( R and S and no T ) - n ( R and T and no S ) - n( R and T and S) = 55 - 10 - 20 - 5 = 20

n ( only S ) = n(S) - n ( R and S and no T ) - n ( S and T and no R ) - n( R and T and S) = 25 - 10 - 5 - 5 = 5

n ( only T ) = n(R) - n ( T and S and no R ) - n ( R and T and no S ) - n( R and T and S) = 65 - 5 - 20 - 5 = 35

Therefore now that we know about every student,

Number of students surveyed who play none of the sports:

= Total number of students surveyed - n(R only ) - n( T only ) - n( S only ) - n ( R and S and no T ) - n ( S and T and no R ) - n ( R and T and no S ) - n( R and T and S)

= 250 - 20 - 5 - 35 - 10 - 5 - 20 - 5 = 250 - 100 = 150

Therefore 150 students play none of the 3 games

b) Number of students who play racquetball and tennis but not squash

= n ( R and T and no S) = 20 as already computed above

Therefore, Number of students who play racquetball and tennis but not squash is 20

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