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Chapter 14: Problem 2

Evaluate the given binomial coefficient. $$\left(\begin{array}{l}7 \\\2\end{array}\right)$$

Short Answer

Expert verified
The binomial coefficient \(\binom{7}{2}\) evaluates to 21.

Step by step solution

01

Compute Factorials

Calculate the necessary factorials:\n- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\) \n- \(2! = 2 \times 1 = 2\) \n- We also need to calculate \((7-2)!\) \n- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
02

Apply the Binomial Coefficient Formula

Substitute the factorials into the binomial coefficient formula: \(\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{5040}{2 \times 120}\)
03

Simplify the result

Calculate the fraction to get the result: \(\binom{7}{2} = \frac{5040}{240} = 21 \)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\)

If \(f(x)=x^{2}+5 x\) and \(g(x)=2 x-3,\) find \(f(g(x))\) and \(g(f(x))\) (Section 8.4, Example 1)

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n}\). Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Simplify: \(\sqrt{28}-3 \sqrt{7}+\sqrt{63}\)

Solve for \(P: A=\frac{P t}{P+t}\)
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