Fourteen numbered balls (1, 2, 3, …, 14) are divided in 3 groups randomly. Find the probability that the sum of the numbers on the balls, in each grou

Question

Fourteen numbered balls (1, 2, 3, …, 14) are divided in 3 groups randomly. Find the probability that the sum of the numbers on the balls, in each grou

Fourteen numbered balls (1, 2, 3, …, 14) are divided in 3 groups randomly. Find the probability that the sum of the numbers on the balls, in each group, is odd.

Monjur Alahi

7 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Each group should have odd numbered balls.
Case I: Two groups have three odd numbered balls and third has only one odd numbered ball.
Number of such cases = `(7!)/((3!)^(2)xx1!xx2!)xx3^(7)` (as even number of balls can go in any group)
Case II: Two groups have one odd numbered ball each and the third group has five odd numbered balls.
Number of such cases = `(7!)/((1!)^(2)xx5! xx 2!)xx3^(7)`
Total number of cases = Number of ways in which three non-empty group can be formed
`=(3^(14)-.^(3)C_(1)2^(14) + .^(3)C_(2))/(3!)`
`therefore` Required probability = `((7!)/(3!)^(2)+(7!)/(5! xx 2!))/(3^(14) - .^(3)C_(1)2^(14)+.^(3)C_(2))xx3!xx3^(7)`