Question:
Two distinct numbers \( x \) and \( y \) are chosen from 1, 2, 3, 4, 5. The probability that the arithmetic mean of \( x \) and \( y \) is an integer is:
Two distinct numbers \( x \) and \( y \) are chosen from 1, 2, 3, 4, 5. The probability that the arithmetic mean of \( x \) and \( y \) is an integer is:
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Sum of two numbers is even if they have the same parity. If you have \( n_e \) even numbers and \( n_o \) odd numbers, the number of favorable pairs is always \( \binom{n_e}{2} + \binom{n_o}{2} \).
Updated On: May 1, 2026
- \( 0 \)
- \( \frac{1}{5} \)
- \( \frac{3}{5} \)
- \( \frac{2}{5} \)
- \( \frac{4}{5} \)
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The Correct Option is D
Solution and Explanation
Concept: The arithmetic mean of two numbers \( (x+y)/2 \) is an integer if and only if their sum \( x+y \) is an even number. For the sum of two integers to be even, both numbers must either be even or both must be odd.
Step 1: Determine the total number of ways to choose two distinct numbers.
We are choosing 2 distinct numbers from a set of 5: {1, 2, 3, 4, 5}. Total outcomes = \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \). The possible pairs are: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5).
Step 2: Identify favorable outcomes (Even Sums).
A pair results in an integer mean if both numbers are odd or both are even.
• Odd numbers available: {1, 3, 5} (3 numbers). Ways to pick 2: \( \binom{3}{2} = 3 \). (Pairs: {1,3}, {1,5}, {3,5}).
• Even numbers available: {2, 4} (2 numbers). Ways to pick 2: \( \binom{2}{2} = 1 \). (Pair: {2,4}). Total favorable outcomes = \( 3 + 1 = 4 \).
Step 3: Calculate the probability.
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{4}{10} = \frac{2}{5} \]
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