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Chapter 10: Problem 58

For small values of \(n\), the binomial coefficient \(\left(\begin{array}{l}n \\\ k\end{array}\right)\) may be quickly found by using Pascal's triangle. To [orm Pascal's triangle, we start with a 1 and place l's diagonally down to the left and to the right. Entries inside of Pascal's triangle are found by adding the two entries immediately above it. For example, in Pascal's triangle below, \(2+1=3\) and \(5+10=15\).Both the row numbering and column numbering start from \(0 .\) For example, the green 1 is in row 4 , column 0 , and the red 15 is in row 6, column 2 . The red 15 is located in the sixth row, second column of Pascal's triangle. Verify that \(\left(\begin{array}{l}6 \\ 2\end{array}\right)=15\).

Short Answer

Expert verified
Both methods confirm \(\binom{6}{2} = 15\).

Step by step solution

01

- Understand the Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements. It can be calculated using Pascal's Triangle or the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
02

- Identify the Row and Column in Pascal's Triangle

In Pascal's Triangle, the row number corresponds to \(n\), and the column number corresponds to \(k\). To verify \(\binom{6}{2} = 15\), locate the 6th row and the 2nd column in Pascal's Triangle.
03

- Construct Pascal's Triangle Up to the Required Row

Construct Pascal's Triangle up to the 6th row:\[\begin{array}{ccccccc} & & & 1 & & & \ & & 1 & & 1 && \ & 1 & & 2 & & 1 & \1 && 3 && 3 && 1 \& 1 & 4 & & 6 & & 4 & & 1 \1 & 5 & & 10 & 10 & 5 & 1 \& 1 & 6 & 15 & 20 & 15 & 6 & 1\end{array}\]
04

- Identify the Value

From the constructed Pascal's Triangle, locate the value in the 6th row and the 2nd column. The value is 15.
05

- Verify Using Binomial Formula

Verify by calculating \(\binom{6}{2}\) using the binomial coefficient formula:\[\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pascal's Triangle
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. This structure is incredibly useful in combinatorial mathematics, especially for finding binomial coefficients. The triangle starts with a single 1 at the top. Each subsequent row starts and ends with 1, and every interior number is the sum of the two numbers above it.
By using Pascal's Triangle, you can quickly find the binomial coefficient \( \binom{n}{k} \) without resorting to complicated calculations. In the triangle, the row number corresponds to n and the column number corresponds to k. So, locating the 6th row and the 2nd column gives us the value 15, which is \( \binom{6}{2} \).
Here鈥檚 how to generate Pascal's Triangle up to the 6th row:
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