Question

The number of integer \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\), such that \(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=20\) and \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \geq 0\) is (a) \({ }^{24} \mathrm{C}_{3}\) (b) \({ }^{25} \mathrm{C}_{3}\) (c) \({ }^{26} \mathrm{C}_{3}\) (d) \({ }^{27} \mathrm{C}_{3}\)

Short answer

The short answer is: (a) \({ }^{24} \mathrm{C}_{3}\).

Step-by-step solution

Stars and Bars Method

To solve this problem, we will use the stars and bars technique. This method is used to count the number of ways to distribute items into bins. In this case, we have 20 items (the sum of a, b, c, and d) and want to distribute them into 4 bins (a, b, c, and d).

Visualizing the Problem

Imagine we have 20 stars representing the sum of a, b, c, and d. We need to place 3 bars in between the stars to separate them into 4 groups, each representing the value of a, b, c, and d respectively. For example, let's suppose the stars and bars look like this: **|***|*****|********* This corresponds to a=2, b=3, c=5, and d=10, since 2+3+5+10=20.

Counting the Ways to Place Bars

Now we need to count the number of ways we can place 3 bars between the 20 stars, keeping in mind that a, b, c, and d can be 0. To do this, we need to consider the 3 bars as separate items along with 20 stars. This gives us 20+3=23 distinct positions. We now have to choose 3 of those 23 positions to place the bars. The number of ways to do this is given by the binomial coefficient: \(\binom{23}{3} \)

Find the Answer and Match with Options

We can now find the value of \(\binom{23}{3}\): \(\binom{23}{3} = \frac{23!}{(23-3)!3!} = \frac{23!}{20!3!} = 1771\) This matches with option (a), so the answer is: (a) \({ }^{24} \mathrm{C}_{3}\)

Key concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns in sets of elements. It provides tools and techniques for solving problems where you have to calculate the number of possible arrangements or groups that can be formed under certain conditions.

For instance, if we consider distributing 20 identical items across four distinct boxes, combinatorics helps us figure out how many different ways this can be done. Such problems often arise in probability, statistics, and many real-world applications such as creating schedules, organizing tournaments, or even in coding and cryptography.

Key Principles in Combinatorics

• Permutation: Arrangement of elements in a definite order.
• Combination: Selection of elements from a larger set where order does not matter.
• Principle of Addition: If an event can occur in 'm' ways and another independent event in 'n' ways, the events together can occur in m+n ways.
• Principle of Multiplication: If an event can occur in 'm' ways and a subsequent independent event in 'n' ways, then the total number of ways the two events can occur is m*n.

Partition of Integers

The partition of integers is a concept in combinatorics where we break down an integer into a sum of other integers. When the integers in the partition are distinct, we refer to it as a partition into distinct parts. However, for our purposes, we usually deal with partitions where repetition is allowed.

Consider we have an integer 'n' and want to express it as a sum of 'k' non-negative parts. Each distinct way in which we can express this sum is called a partition. For example, the number 4 can be partitioned into 1+1+1+1, 2+1+1, 2+2, 3+1, and 4 (if considering only up to four parts).

The stars and bars method is a graphical technique used in combinatorics which simplifies the process of finding partitions. By representing the parts as 'stars' and the dividers between them as 'bars', we can easily visualize and count the number of partitions of an integer into a specific number of parts.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, represented by the notation \(\binom{n}{k}\) or sometimes as \(nCk\). It captures the idea of 'selection' or 'combination' and is used to calculate the number of ways to choose 'k' elements from a larger set of 'n' elements without regard to the order of selection.

The mathematical formula for the binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) where 'n!' denotes the factorial of 'n', which is the product of all positive integers up to 'n'.

For example, the solution to our exercise involved the binomial coefficient \(\binom{23}{3}\), representing the number of ways to place 3 bars among 23 positions. This aligns with the stars and bars method by providing an efficient way to count partitions of an integer (in our case, the number 20) into a sum of a fixed number of terms (4 terms a, b, c, and d).