Question

If \(m\) things are distributed among \(a\) men and \(b\) women, then the chance that the number of things received by men is odd is:

• A. \(\frac{(b-a)^m - (b+a)^m}{2(b+a)^m}\)
• B. \(\frac{(b+a)^m - (b-a)^m}{2(b+a)^m}\)
• C. \(\frac{(b+a)^m - (b-a)^m}{(b+a)^m}\)
• D. None of these

Hint:

Odd-even separation uses \((x+y)^n \pm (x-y)^n\).

Answer 1

The Correct Option is B

Concept: Each object has \((a+b)\) choices.

Step 1: Total ways.
\[ (a+b)^m \]

Step 2: Use binomial expansion.
Number of ways men get odd objects: \[ \frac{(a+b)^m - (b-a)^m}{2} \]

Step 3: Probability.
\[ \frac{(a+b)^m - (b-a)^m}{2(a+b)^m} \]