<p>Fourteen numbered balls \((1, 2, 3, \ldots, 14)\) are divided in 3 groups randomly. Find the probability that the sum of the numbers on the balls, in each group, is odd.</p>
<p>\(\dfrac{\dfrac{7!}{(3!)^2 \times 1! \times 2!} \times 3^7 + \dfrac{7!}{(1!)^2 \times 5! \times 2!} \times 3^7}{3^{14} - {}^3C_1 2^{14} + {}^3C_2}\)</p>
<p>\(\dfrac{\dfrac{7!}{(3!)^2} + \dfrac{7!}{5!}}{3^{14}}\)</p>
<p>\(\dfrac{3^7}{3^{14} - {}^3C_1 2^{14} + {}^3C_2}\)</p>
<p>\(\dfrac{\dfrac{7!}{(1!)^2 \times 5! \times 2!}}{3^{14} - {}^3C_1 2^{14} + {}^3C_2}\)</p>
Step-by-Step Solution
Key Concept: For a sum to be odd, a group must contain an odd number of odd-numbered balls. Since there are 7 odd and 7 even balls total, we need each of the 3 groups to have odd count of odd balls, which constrains the distribution to (1,3,3) or (3,1,3) or (3,3,1) odd balls per group.
<p><strong>Step 1: Identify parity condition</strong></p><p>For a group's sum to be odd, it must contain an ODD number of odd-numbered balls. (Even balls don't affect parity.)</p><p><strong>Step 2: Odd ball distribution</strong></p><p>We have 7 odd-numbered balls (1,3,5,7,9,11,13) and 7 even-numbered balls (2,4,6,8,10,12,14). For all three groups to have odd sums, each must contain an odd number of odd balls.</p><p><strong>Step 3: Find valid partitions</strong></p><p>We need to partition 7 into three positive odd numbers: 7 = 1+3+3 (and permutations). This is the ONLY way.</p><p><strong>Step 4: Count favorable outcomes</strong></p><p>Number of ways to distribute 7 odd balls as (1,3,3): <strong>C(7,1)·C(6,3)·C(3,3) = 7·20·1 = 140</strong></p><p>Number of permutations of groups (1,3,3): <strong>3!/2! = 3</strong></p><p>Total favorable ways to distribute odd balls: <strong>140·3 = 420</strong></p><p>Even balls can be distributed freely among 3 groups: <strong>3^7 ways</strong></p><p>Favorable outcomes: <strong>420·3^7</strong></p><p><strong>Step 5: Total outcomes and probability</strong></p><p>Total ways to divide 14 balls into 3 groups: <strong>3^14</strong></p><p>Probability = (420·3^7)/3^14 = 420/3^7 = 420/2187 = <strong>140/729</strong></p><p>∴ Answer: A</p>
Correct Answer: A