/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Evaluate the given binomial coef... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 8

Evaluate the given binomial coefficient. $$\left(\begin{array}{c}100 \\\98\end{array}\right)$$

Short Answer

Expert verified
4950

Step by step solution

01

Substitute into the formula

Substitute n = 100 and k = 98 into the formula for the binomial coefficient: \[ \left(\begin{array}{c}n \k\end{array}\right) = \frac{n!}{k!(n-k)!} \] This gives \[\left(\begin{array}{c}100 \98\end{array}\right) = \frac{100!}{98!(100-98)!} \] which simplifies to \[\frac{100!}{98!2!}\]
02

Simplify the factorial

Simplify the factorial, where \( n! = n * (n-1)! \). That is, \( 100! = 100 * 99 * 98! \) and also \( 2! = 2 * 1! \). So the expression can be simplified to: \[\frac{100 * 99 * 98!}{98! * 2 * 1}\]
03

Cancel common factors

The common factors in the numerator and denominator can be cancelled, since \( 98! / 98! = 1 \) . After cancelling out common factors, the expression simplifies to: \[\frac{100 * 99 }{2 * 1}\] or \[50 * 99 = 4950\]
04

Check if you can simplify further

You cannot simplify this expression any further.

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