If H is the Harmonic Mean of three numbers {}^{10}C_{4}, {}^{10}C_{5}, and {}^{10}C_{6}, then what is the value of \frac{270}{H}?
- A. 1
- B. \frac{14}{17}
- C. \frac{17}{14} ✓
- D. \frac{1}{31}
Correct Answer: C. \frac{17}{14}
Explanation
Calculate the values: {}^{10}C_4 = 210, {}^{10}C_5 = 252, and {}^{10}C_6 = 210. The harmonic mean H of three numbers a, b, c is H = \frac{3}{1/a + 1/b + 1/c}. Substituting the values gives H = \frac{3}{2/210 + 1/252} = \frac{3}{1/105 + 1/252}. The LCM of 105 and 252 is 1260. This yields H = \frac{3}{(12+5)/1260} = \frac{3780}{17}. We need \frac{270}{H} = \frac{270 \times 17}{3780} = \frac{17}{14}.
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