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Chapter 14: Problem 6

Evaluate the given binomial coefficient. $$\left(\begin{array}{c}15 \\\2\end{array}\right)$$

Short Answer

Expert verified
The value of the binomial coefficient is 105.

Step by step solution

01

Understand The Binomial Coefficient

The binomial coefficient is a mathematical way to determine the number of ways to choose a subset of items without regard to order. It is given by the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) means 'n factorial', which is the product of all positive integers up to \(n\).
02

Apply the values into the formula

Now we put our given values \(n = 15\) and \(k = 2\) into the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). This gives us \(\binom{15}{2} = \frac{15!}{2!(15-2)!}\).
03

Simplify the values

First, calculate the factorials and the subtraction in the denominator: \(15! = 1*2*3...*15\), \(2! = 2\) and \(15-2 = 13\), so \(13! = 1*2*3...*13\). Now we can simplify our expression to: \(\binom{15}{2} = \frac{15!}{2!*13!}\).
04

Cancel unnecessary terms

Notice that \(15!\) in the numerator includes terms from \(1*2*3...*13*14*15\) and \(13!\) in the denominator has terms from \(1*2*3...*13\). We can cancel out terms from 1 to 13 in both numerator and denominator . Now we are left with \(\binom{15}{2} = \frac{14*15}{2}\) in simpler form.
05

Calculate the final answer

Finally, perform the remaining multiplication and division to get the binomial coefficient value. Calculation gives us \(\binom{15}{2} = 105\).

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(10-5+\frac{5}{2}-\frac{5}{4}+\cdots=\frac{10}{1-\frac{1}{2}}\)

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{n}{n+1} ; n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

Use the formula for the sum of an infinite geometric series to solve Exercises. A new factory in a small town has an annual payroll of \(\$ 6\) million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

Use the formula for the sum of the first n terms of a geometric sequence to solve. You are investigating two employment opportunities. Company A offers \(\$ 30,000\) the first year. During the next four years, the salary is guaranteed to increase by \(6 \%\) per year. Company B offers \(\$ 32,000\) the first year, with guaranteed annual increases of \(3 \%\) per year after that. Which company offers the better total salary for a five-year contract? By how much? Round to the nearest dollar.
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