QJEEM24Ma23b7: JEE Main 2024 Mathematics - Class 11 JEEM24Ma23b7 Solution: The number of ways in which 21 identical apples can be…
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
Multiple Choice Options
- A 130
- B 136
- C 142
- D 406
Step-by-Step Verified Solution
To solve this problem, we can use a classic combinatorics method known as "stars and bars" (or "balls and bins"), which is a way to solve problems involving distributing identical items into distinct groups with certain restrictions.
First, since each child must get at least 2 apples, let's give 2 apples to each child right away. That accounts for 6 apples (2 apples for each of the 3 children). Now, we have 21 - 6 = 15 apples left to distribute freely among the three children.
The "stars and bars" technique involves representing the apples as stars (*) and the divisions between children as bars (|). For example, if we had 5 apples to distribute among three children, one possible distribution could be represented as **|*|**. This means the first child gets 2 apples, the second child gets 1 apple, and the third child gets 2 apples.
In our case, we need to distribute 15 apples (stars) among the three children with 2 bars to create the partitions. We arrange 15 stars and 2 bars in a row, where the arrangement of stars and bars corresponds to a distribution of the apples.
The total number of objects we're arranging is 15 apples + 2 bars = 17 objects. We need to choose 2 positions out of these 17 to place the bars. The remaining positions will be occupied by the stars (apples).
The number of ways to choose 2 positions out of 17 for the bars is given by the binomial coefficient:
Thus, there are 136 ways to distribute the 21 identical apples among three children such that each child gets at least 2 apples. The correct answer is Option B: 136.