NIMCET 2008 — Mathematics PYQ

The condition $f(i)\le f(j)$ for i < j means the function must be non-decreasing.

• Set $A$ (Domain) has $3$ elements: $\{0,1,2\}$.

• Set $B$ (Codomain) has $8$ elements: $\{0,1,2,3,4,5,6,7\}$.

We need to choose $3$ values from set $B$ to map to the $3$ elements in set $A$. Because the function must be non-decreasing, once we pick any $3$ values (allowing repetitions), there is only one way to arrange them to satisfy $f(0)\le f(1)\le f(2)$.

Step 2: Apply the Formula for Combinations with Repetition

Finding the number of non-decreasing functions is equivalent to selecting $r$ items from $n$ items where repetition is allowed and order does not matter.

• $n=|B|=8$ (Available choices)

• $r=|A|=3$ (Number of selections needed)

The formula for combinations with repetition (also known as stars and bars) is:

Step 3: Substitute the Values

Substitute $n=8$ and $r=3$:

Step 4: Calculate the Value

Final Answer:

The total number of such functions is ${}^{10}{C}_3$.

The correct option is (c).