Solution: 2015 Fall Midterm - 3
Author: Michiel SmidQuestion
Let $A$ be a set of 6 elements and let $B$ be a set of 13 elements. How many one-to-one (i.e.,
injective) functions $f : A \rightarrow B$ are there?
(a)
$8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$
(b)
$6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$
(c)
$7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$
(d)
$5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$
Solution
The first element in A can be mapped to any of the 13 elements in B = 13 possibilities
The second element in A can be mapped to any of the 12 remaining elements in B = 12 possibilities
The third element in A can be mapped to any of the 11 remaining elements in B = 11 possibilities
The fourth element in A can be mapped to any of the 10 remaining elements in B = 10 possibilities
The fifth element in A can be mapped to any of the 9 remaining elements in B = 9 possibilities
The sixth element in A can be mapped to any of the 8 remaining elements in B = 8 possibilities
Thus, the number of one-to-one functions is $13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8$