First slide

Combinations

Question

The number of functions $f$ from the set A $= {0, 1, 2}$ in to the set $B = {0, 1, 2, 3, 4, 5, 6, 7}$ such that $f (i) \leq f (j) for i < j$ and $i, j \in A$ is

Moderate

A

$^{8} C_{3}$
B

$^{8} C_{3} + 2 (^{8} C_{2})$
C

$^{10} C_{3}$
D

$^{10} C_{4}$

Solution

A function $f : A \to B$ such that $f (0) \leq f (1) \leq$ $f (2)$ falls in one of the following four categories.
Case 1 $f (0) < f (1) < f (2)$
There are $^{8} C_{3}$ functions in this category.
Case 2 $f (0) = f (1) < f (2)$
There are $^{8} C_{2}$ functions in this category
Case 3 $f (0) < f (1) = f (2)$
There are again $^{8} C_{2}$ functions in this category
Case 4 $f (0) = f (1) = f (2)$
There are $^{8} C_{1}$ functions in this category. Thus, the number of desired functions is
$^{8} C_{3} +^{8} C_{2} +^{8} C_{2} +^{8} C_{1} =^{9} C_{3} +^{9} C_{2} =^{10} C_{3}$ .

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