Question

Number of ways of distributing 16 identical oranges among 4 persons such that each one gets at least one orange is:

• A. 435
• B. 455
• C. 470
• D. 489

Hint:

For distribution of identical objects with minimum constraints, first satisfy the minimum condition and then apply the stars and bars method.

Answer 1

The Correct Option is B

To solve the given problem of distributing 16 identical oranges among 4 persons such that each person receives at least one orange, we can use the combinatorial method known as "Stars and Bars". This method allows us to distribute items (oranges) into groups (people) with the condition that each group gets at least one item. 

The general formula for distributing \( n \) identical items into \( r \) groups where each group receives at least one item is given by:

\(n = k + r - 1\)

where:

• \(n\) is the total number of identical items.
• \(k\) is the number of groups.

 

To ensure that each group gets at least one item, we first distribute 1 item to each group. This is because the question specifies that each person must get at least one orange. Therefore, we have already used 4 oranges (since there are 4 people).

Now, we must distribute the remaining oranges:

Total oranges to be distributed = 16 - 4 = 12 oranges

We need to find the number of non-negative integer solutions to the equation:

\(x_1 + x_2 + x_3 + x_4 = 12\)

where \(x_1, x_2, x_3,\) and \(x_4\) represent the number of additional oranges each person can receive beyond the initial one.

Using the "Stars and Bars" theorem, the number of ways to solve the equation is given by:

\(\binom{n+k-1}{k-1} = \binom{12+4-1}{4-1} = \binom{15}{3}\)

Calculating the binomial coefficient:

\(\binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455\)

Therefore, the number of ways to distribute the 16 identical oranges among 4 persons such that each person gets at least one orange is 455.