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Chapter 34: Chapter 34

Problem 1076

Next question

Evaluate each of the following symbols: (a) \(\mathrm{C}(6,3)\) (b) \(\mathrm{C}(18,16)\)

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\(\mathrm{C}(6,3) = 20\) and \(\mathrm{C}(18,16) = 153\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: (Part (a))

We need to evaluate C(6, 3) using the formula: \[\mathrm{C}(6, 3) = \frac{6!}{3!(6-3)!}\] First, let's calculate the factorials: - 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 - 3! = 3 × 2 × 1 = 6 - (6-3)! = 3! = 6 Now, substitute the factorials into the formula and calculate the value: \[\mathrm{C}(6, 3) = \frac{720}{6 \times 6} = \frac{720}{36} = 20\] So, \(\mathrm{C}(6,3) = 20\).

Step 2: (Part (b))

We need to evaluate C(18, 16) using the formula: \[\mathrm{C}(18, 16) = \frac{18!}{16!(18-16)!}\] First, let's calculate the factorials: - 18! = 18 × 17 × 16 × ... × 3 × 2 × 1 = 6,402,373,705,728,000 - 16! = 16 × 15 × 14 × ... × 3 × 2 × 1 = 20,922,789,888,000 - (18-16)! = 2! = 2 × 1 = 2 Now, substitute the factorials into the formula and calculate the value: \[\mathrm{C}(18, 16) = \frac{6,402,373,705,728,000}{20,922,789,888,000 \times 2} = \frac{6,402,373,705,728,000}{41,845,579,776,000} = 153\] So, \(\mathrm{C}(18,16) = 153\).

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