Question

Consider all functions that can be defined from the set A = {1,2,3} to set B = {1,2,3,4,5,6,7} and a function f(x) is selected at random from these functions. If the probability that selected function satisfies f(i) ≤ f(j) for i<j is equal to p, then greatest integral value of $\frac{1}{p}$ is equal to:

Answer 1

4

Answer 2

Here's how to solve this problem:

1. Total number of functions: Since a function $f : A \rightarrow B$ with $|A|=3$ and $|B|=7$ can be any assignment, the total number of functions is

   $$7^3 = 343.$$

2. Counting non-decreasing functions: We require $f(1) \le f(2) \le f(3)$. This is equivalent to choosing a multiset of 3 elements from 7 available elements. The number of ways is given by the “combination with repetition” formula:

   $$\binom{7+3-1}{3} = \binom{9}{3} = 84.$$

3. Probability $p$:

   $$p = \frac{84}{343}.$$

4. Compute $\frac{1}{p}$:

   $$\frac{1}{p} = \frac{343}{84} = \frac{7^3}{7 \times 12} = \frac{49}{12} \approx 4.0833.$$

   The greatest integral value (i.e. the floor) is 4.