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Chapter 9: Problem 17

Using Pascal's Triangle In Exercises 15-18, evaluate using Pascal's Triangle. $$_{7} C_{4}$$

Short Answer

Expert verified
The binomial coefficient \(_{7} C_{4}\) equals 35.

Step by step solution

01

Understanding Pascal's Triangle

In Pascal's row, the topmost row is row 0 containing only 1. Then the numbers in the following rows are generated by adding the number above and to its left and the number above and to its right from the previous row. If there's no number in the spot, it is treated as zero.
02

Counting the Rows till 7

Start counting the rows from 0 till we reach row 7. Row 7 of Pascal's Triangle ends up being: 1, 7, 21, 35, 35, 21, 7, 1.
03

Finding \(_{7} C_{4}\)

The binomial coefficient \(_{7} C_{4}\) is the 5th term in row 7 (keeping in mind that we start counting from 0). Hence \(_{7} C_{4}\) is equal to 35.

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Key Concepts

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

Binomial Coefficient
A binomial coefficient is an important number in mathematics, especially in algebra, and it is heavily used when working with combinations and binomial expansions.
The binomial coefficient is expressed as \( \binom{n}{k} \), which represents the number of ways to choose \( k \) elements from \( n \) elements without regard to order.
This is why it is often called "n choose k".
  • In mathematics, the formula to calculate the binomial coefficient is given by:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \( n! \) denotes factorial, which is the product of all positive integers up to \( n \).
  • For example, to calculate \( \binom{7}{4} \), apply the formula:\[\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1 \cdot 3 \times 2 \times 1} = 35\]
In this case, the result is consistent with what we find in Pascal's Triangle.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting.
It explores how objects can be arranged and counted in various scenarios.
Think about it like organizing, counting, and choosing different groups of items.
  • It's used in various fields like computer science, statistics, and optimization.
  • In the context of binomial coefficients, combinatorics helps us understand how to count and choose elements efficiently.
  • The binomial coefficient \( \binom{n}{k} \) from combinatorics tells us the number of ways to pick \( k \) items from \( n \) total items, which aligns with the idea of forming and organizing groups.
The beauty of combinatorics is in its versatility and frequent application in solving real-world problems.
Row 7 in Pascal's Triangle
Pascal's Triangle is a powerful and simple tool for visualizing binomial coefficients.
Each row in Pascal's Triangle corresponds to a power of a binomial (a sum of two terms).
Let's explore Row 7.
  • Counting starts at row 0 at the top. Row 7 is the 8th from the top because counting starts at zero.
  • The entries in Row 7 are 1, 7, 21, 35, 35, 21, 7, 1.
  • The 5th entry (counting from 0) of Row 7 gives us the binomial coefficient \(_{7} C_{4} \), which is 35.
Pascal's Triangle demonstrates the symmetry in binomial coefficients, which parallels the concepts in combinatorics effectively.

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Most popular questions from this chapter

In Exercises \(75-82,\) solve for \(n\) $$_{n} P_{5}=18 \cdot_{n-2} P_{4}$$

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{30} n$$

You are given the probability that an event will happen. Find the probability the event will not happen. \(P(E)=0.36\)

Forming Rows of Pascal's Triangle Form rows \(8-10\) of Pascal's Triangle.

Finding a Quadratic Model In Exercises \(69-74\) find a quadratic model for the sequence with the indicated terms. $$a_{0}=3, a_{1}=3, a_{4}=15$$
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