Problem 119
Question
In how many ways can 20 oranges be given to four children if each child should get at least one orange? (A) 869 (B) 969 (C) 973 (D) None of these
Step-by-Step Solution
Verified Answer
There are 969 ways to distribute the oranges.
1Step 1: Understanding the Problem
We need to determine how to distribute 20 oranges among 4 children such that each child receives at least one orange. This initial requirement means each child must get at least 1 orange, leaving us to distribute the rest.
2Step 2: Distributing Minimum Oranges
Distribute 1 orange to each of the 4 children initially, which accounts for 4 oranges. Now, we have 20 - 4 = 16 oranges left to distribute.
3Step 3: Using Stars and Bars Technique
The remaining 16 oranges must now be distributed among 4 children, where a child can receive 0 or more oranges. We use the stars and bars method, where the formula to find the number of distributions is given by choosing \(k - 1\) bars (dividers) from \(n + k - 1\) total items (stars + bars). Here, \(n = 16\) and \(k = 4\).
4Step 4: Apply the Formula
The formula is \(\binom{n + k - 1}{k - 1}\). So we calculate \(\binom{16 + 4 - 1}{4 - 1} = \binom{19}{3}\).
5Step 5: Calculate the Combinatorics
Calculate \(\binom{19}{3}\) which is \(\frac{19 \times 18 \times 17}{3 \times 2 \times 1} = 969\).
6Step 6: Choose the Correct Option
The answer corresponds to option (B) 969 from the given choices.
Key Concepts
Stars and Bars MethodDistribution ProblemsBinomial Coefficient
Stars and Bars Method
The Stars and Bars method is a clever combinatorial technique used to solve distribution problems. It helps determine the number of ways to distribute identical items into distinct groups. In our exercise's context, we're distributing 20 oranges among 4 children.
First, you give each child one orange to ensure they all receive at least one. This leaves 16 oranges to be distributed freely. The core idea of Stars and Bars is to consider these remaining oranges as 'stars'. You then need 'bars' to separate these stars among the children.
In general, if you have 'n' items (like oranges) and 'k' groups (like children), the formula you use is a combination formula, typically written as \(\binom{n + k - 1}{k - 1}\). This formula calculates the number of ways to insert the 'bars' among the 'stars'. Here, each arrangement of stars and bars represents a valid distribution of oranges.
First, you give each child one orange to ensure they all receive at least one. This leaves 16 oranges to be distributed freely. The core idea of Stars and Bars is to consider these remaining oranges as 'stars'. You then need 'bars' to separate these stars among the children.
In general, if you have 'n' items (like oranges) and 'k' groups (like children), the formula you use is a combination formula, typically written as \(\binom{n + k - 1}{k - 1}\). This formula calculates the number of ways to insert the 'bars' among the 'stars'. Here, each arrangement of stars and bars represents a valid distribution of oranges.
Distribution Problems
Distribution problems typically involve finding ways to allocate a certain number of identical items to different recipients. They are very common in combinatorics where constraints, such as each recipient getting at least one item, must be taken into account.
In our context, the problem is to give 20 oranges to four children such that each child gets at least one orange. To handle this, you start by ensuring constraints are met – giving each child 1 orange to start with. After fulfilling this condition, problems usually reduce to finding the number of unrestricted ways to distribute the remaining items using methods like Stars and Bars.
These problems often require setting up an equation where total items minus what each recipient already has are distributed freely using combinatorial principles. It's essential to recognize how initial constraints transform a problem into a simpler form.
In our context, the problem is to give 20 oranges to four children such that each child gets at least one orange. To handle this, you start by ensuring constraints are met – giving each child 1 orange to start with. After fulfilling this condition, problems usually reduce to finding the number of unrestricted ways to distribute the remaining items using methods like Stars and Bars.
These problems often require setting up an equation where total items minus what each recipient already has are distributed freely using combinatorial principles. It's essential to recognize how initial constraints transform a problem into a simpler form.
Binomial Coefficient
The Binomial Coefficient is a crucial mathematical tool used in combinations and permutations, essential for solving various distribution problems. It's denoted as \(\binom{n}{r}\) and represents the number of ways to choose 'r' elements from a set of 'n' elements without considering the order.
In our problem, once we've given each child one orange and are left with 16 to distribute, we use the formula \(\binom{n + k - 1}{k - 1}\) from Stars and Bars. Specifically, this becomes \(\binom{19}{3}\), representing the number of ways to position 3 dividers (bars) among 19 positions (16 stars + 3 bars) to partition the oranges among 4 children.
Calculating \(\binom{19}{3}\) involves simple arithmetic: \(\frac{19 \times 18 \times 17}{3 \times 2 \times 1}\), which equals to 969. The Binomial Coefficient thus helps solve how the remaining items can be distributed effectively while considering constraints.
In our problem, once we've given each child one orange and are left with 16 to distribute, we use the formula \(\binom{n + k - 1}{k - 1}\) from Stars and Bars. Specifically, this becomes \(\binom{19}{3}\), representing the number of ways to position 3 dividers (bars) among 19 positions (16 stars + 3 bars) to partition the oranges among 4 children.
Calculating \(\binom{19}{3}\) involves simple arithmetic: \(\frac{19 \times 18 \times 17}{3 \times 2 \times 1}\), which equals to 969. The Binomial Coefficient thus helps solve how the remaining items can be distributed effectively while considering constraints.
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