First slide

Functions (XII)

Question

$L e t X = \{a_{1}, a_{2}, . . . ., a_{6}\} a n d Y = \{b_{1}, b_{2}, b_{3}\} .$ The number of functions f from X to Y such that it is onto and there are exactly three elements x in X such that $f (x) = b_{1}$ is

Moderate

A

75
B

90
C

100
D

120

Solution

Image $b_{1}$ is assigned to any three of the six pre-images in ${}^{6}{C_{3}}$ ways.
Rest two images can be assigned to remaining three pre-images in $2^{3} - 2$ ways (as function is onto).
Hence, number of functions are ${}^{6}{C_{3}} \times (2^{3} - 2) = 20 \times 6 = 120$ .

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