To find the largest and smallest possible values for the expression $(x \times y) + z$, we must choose distinct values for $x$, $y$, and $z$ from the given set $\{2, 3, 4, 5\}$.

Step 1: Finding the largest possible value ($M$) To maximize the value of $(x \times y) + z$, we should maximize the product $(x \times y)$ because multiplication of numbers greater than $1$ yields a larger result than addition.

• The two largest numbers available are $4$ and $5$. Therefore, we assign $x = 4$ and $y = 5$ (or vice versa), giving a product of $4 \times 5 = 20$.
• The remaining numbers are $2$ and $3$. To maximize the overall sum, we assign the largest remaining number to $z$, which is $3$.
• Calculating the maximum value: $M = (4 \times 5) + 3 = 20 + 3 = 23$

Step 2: Finding the smallest possible value ($N$) To minimize the value of $(x \times y) + z$, we should minimize the product $(x \times y)$.

• The two smallest numbers available are $2$ and $3$. Therefore, we assign $x = 2$ and $y = 3$ (or vice versa), giving a product of $2 \times 3 = 6$.
• The remaining numbers are $4$ and $5$. To minimize the overall sum, we assign the smallest remaining number to $z$, which is $4$.
• Calculating the minimum value: $N = (2 \times 3) + 4 = 6 + 4 = 10$

(Self-check: If we tried $x=2, y=4, z=3$, the result would be $(2 \times 4) + 3 = 11$, which is larger than $10$. Thus, $10$ is indeed the absolute minimum.)

Step 3: Finding the difference ($M - N$) Now, we subtract the smallest value from the largest value: $M - N = 23 - 10 = 13$

Therefore, the difference between the largest and smallest possible values is $13$.