Step-by-step Solution:

1. Identify the given information: Let X be a set with 5 elements, so $|X| = 5$. Let Y be a set with 7 elements, so $|Y| = 7$.

2. Calculate α, the number of one-one functions from X to Y. A one-one (injective) function from X to Y maps each element of X to a unique element in Y. To construct such a function, we choose images for the 5 elements of X from the 7 elements of Y without replacement.

• The first element of X can be mapped to any of the 7 elements in Y.
• The second element of X can be mapped to any of the remaining 6 elements in Y.
• The third element of X can be mapped to any of the remaining 5 elements in Y.
• The fourth element of X can be mapped to any of the remaining 4 elements in Y.
• The fifth element of X can be mapped to any of the remaining 3 elements in Y.

So, the total number of one-one functions, α, is the number of permutations of 7 items taken 5 at a time, denoted as $^7P_5$. $\alpha = ^7P_5 = 7 \times 6 \times 5 \times 4 \times 3$ $\alpha = 2520$

3. Calculate β, the number of onto functions from Y to X. An onto (surjective) function from Y to X maps elements of Y to X such that every element in X is the image of at least one element in Y. The domain is Y with $|Y|=7$ elements, and the codomain is X with $|X|=5$ elements.

The number of onto functions from a set of size $m$ to a set of size $n$ is given by the formula using the principle of inclusion-exclusion: $\text{Number of onto functions} = \sum_{k=0}^{n} (-1)^k \, ^nC_k \, (n-k)^m$ Here, $m=7$ and $n=5$. $\beta = \sum_{k=0}^{5} (-1)^k \, ^5C_k \, (5-k)^7$ Expanding the sum: $\beta = ^5C_0(5-0)^7 - ^5C_1(5-1)^7 + ^5C_2(5-2)^7 - ^5C_3(5-3)^7 + ^5C_4(5-4)^7 - ^5C_5(5-5)^7$ $\beta = ^5C_0 \cdot 5^7 - ^5C_1 \cdot 4^7 + ^5C_2 \cdot 3^7 - ^5C_3 \cdot 2^7 + ^5C_4 \cdot 1^7 - ^5C_5 \cdot 0^7$ Now, we calculate the values of the terms:

• $^5C_0 = 1$, $5^7 = 78125$
• $^5C_1 = 5$, $4^7 = 16384$
• $^5C_2 = 10$, $3^7 = 2187$
• $^5C_3 = 10$, $2^7 = 128$
• $^5C_4 = 5$, $1^7 = 1$
• $^5C_5 = 1$, $0^7 = 0$

Substitute these values back into the expression for β: $\beta = (1)(78125) - (5)(16384) + (10)(2187) - (10)(128) + (5)(1) - (1)(0)$ $\beta = 78125 - 81920 + 21870 - 1280 + 5$ $\beta = (78125 + 21870 + 5) - (81920 + 1280)$ $\beta = 100000 - 83200$ $\beta = 16800$

4. Calculate the value of the expression ${1 \over {5!}}(\beta - \alpha )$. First, calculate $5!$: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ Next, calculate $\beta - \alpha$: $\beta - \alpha = 16800 - 2520 = 14280$ Finally, substitute these values into the given expression: ${1 \over {5!}}(\beta - \alpha ) = {1 \over {120}}(14280)$ $= {14280 \over 120} = {1428 \over 12}$ Performing the division: ${1428 \over 12} = 119$

Conclusion: The value of ${1 \over {5!}}(\beta - \alpha )$ is 119.