The correct option is (c) - 13.
[as per provisional answerkey]Solution
We are given three distinct variables x, y, and z from the set {2, 3, 4, 5}. We need to find the difference between the maximum value (M) and the minimum value (N) of the expression {(x×y)+z}.
1. Finding the Maximum Value (M):
To maximize (x×y)+z, we should assign the largest available numbers to the product (x * y) because multiplication scales faster than addition.
Let x = 5 and y = 4 (the two largest numbers).
Then z must be the next largest available number, which is 3.
M=(5×4)+3=20+3=23.
Note: If we chose z = 5, then x and y would be 4 and 3, giving (4×3)+5=17, which is smaller.
2. Finding the Minimum Value (N):
To minimize (x×y)+z, we should assign the smallest available numbers to the product (x * y).
Let x = 2 and y = 3 (the two smallest numbers).
Then z must be the next smallest available number, which is 4.
N=(2×3)+4=6+4=10.
Note: If we chose z = 2, then x and y would be 3 and 4, giving (3×4)+2=14, which is larger.
3. Calculating M−N:
M−N=23−10=13.
Why the other options are incorrect
- Option (a) - 11: This result might be reached if one incorrectly calculates M as 22 or N as 12, failing to optimize the placement of the largest/smallest digits in the product vs. the sum.
- Option (b) - 12: This is a common error if the student assumes the difference between the highest possible product (20) and lowest possible product (6) is the only factor, or makes a minor subtraction error.
- Option (d) - 14: This value exceeds the actual range; it could occur if the distinctness constraint is ignored (e.g., using 5, 5, 4 for M or 2, 2, 3 for N), which violates the rule that values must be distinct.
Key Concept
To maximize an expression of the form (x×y)+z, assign the largest values to the multiplicative terms; to minimize it, assign the smallest values to the multiplicative terms.