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Problem 1076

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

Expert verified
$$\mathrm{C}(6,3) = 20$$ and $$\mathrm{C}(18,16) = 153$$.
See the step by step solution

## 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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