<p>Number of ways of voting = \({}^{10}C_1 + {}^{10}C_2 + {}^{10}C_3 + {}^{10}C_4\). The number of ways is:</p>
Step-by-Step Solution
Key Concept: We need to calculate the sum of binomial coefficients C(10,1) + C(10,2) + C(10,3) + C(10,4) by directly computing each term using the binomial coefficient formula C(n,r) = n!/(r!(n-r)!).
<p><strong>Step 1:</strong> Calculate $\binom{10}{1}$</p><p>$\binom{10}{1} = \frac{10!}{1! \cdot 9!} = 10$</p><p><strong>Step 2:</strong> Calculate $\binom{10}{2}$</p><p>$\binom{10}{2} = \frac{10!}{2! \cdot 8!} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45$</p><p><strong>Step 3:</strong> Calculate $\binom{10}{3}$</p><p>$\binom{10}{3} = \frac{10!}{3! \cdot 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120$</p><p><strong>Step 4:</strong> Calculate $\binom{10}{4}$</p><p>$\binom{10}{4} = \frac{10!}{4! \cdot 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210$</p><p><strong>Step 5:</strong> Add all terms together</p><p>$\binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \binom{10}{4} = 10 + 45 + 120 + 210 = 385$</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C