Question

Number of ways of distributing \(10\) identical chocolates among \(3\) children such that everyone gets at least one?

• A. \(2^4 + 3\cdot2^5 - 2\)
• B. \(2^4 - 3\cdot2^5 - 2\)
• C. \(2^5 + 4\)
• D. \(2^5 - 4\)
• E.

Hint:

For distributing identical objects where each recipient must get at least one, the formula is \(\binom{n-1}{r-1}\). If recipients can receive zero, the formula is \(\binom{n+r-1}{r-1}\).

Answer 1

The Correct Option is C

Concept: When distributing identical objects among distinct people with each receiving at least one object, we use the stars and bars method. The number of ways to distribute \(n\) identical objects among \(r\) persons such that each gets at least one is: \[ \binom{n-1}{r-1} \]

Step 1: Convert the condition into an equation. Let the chocolates received by the three children be \(x_1, x_2, \text{ and } x_3\): \[ x_1 + x_2 + x_3 = 10 \] with the condition that each child receives at least one chocolate: \[ x_1, x_2, x_3 \ge 1 \]

Step 2: Apply the stars and bars formula. Using \(n = 10\) and \(r = 3\): \[ \text{Number of ways} = \binom{10-1}{3-1} = \binom{9}{2} \] \[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

Step 3: Match with the given options. We evaluate the provided expressions:
• (A) \(16 + 3(32) - 2 = 16 + 96 - 2 = 110\)
• (B) \(16 - 96 - 2 = -82\)
• (C) \(2^5 + 4 = 32 + 4 = 36\)
• (D) \(2^5 - 4 = 32 - 4 = 28\) The value \(36\) matches exactly with option (C).