Calculating Harmonic Mean of Binomial Coefficients

The question asks for the value of \( \frac{270}{H} \), where H is the Harmonic Mean of three specific binomial coefficients: \(10C4\), \(10C5\), and \(10C6\).

Step 1: Calculate the Binomial Coefficients

First, we need to calculate the values of the given binomial coefficients. The formula for a binomial coefficient \(nCr\) is given by \(nCr = \frac{n!}{r!(n-r)!}\).

• Calculate \(10C4\): \(10C4 = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4 \times 3 \times 2 \times 1 \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}\) \(10C4 = \frac{5040}{24} = 210\)
• Calculate \(10C5\): \(10C5 = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5 \times 4 \times 3 \times 2 \times 1 \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1}\) \(10C5 = \frac{30240}{120} = 252\)
• Calculate \(10C6\): \(10C6 = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6! \times 4 \times 3 \times 2 \times 1} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}\) \(10C6 = \frac{5040}{24} = 210\)

The three numbers are 210, 252, and 210.

Step 2: Understand the Harmonic Mean Formula

The Harmonic Mean (H) of three numbers \(a, b, c\) is given by the formula:

\(H = \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}\)

In this case, the numbers are \(a = 210\), \(b = 252\), and \(c = 210\).

Step 3: Calculate the Harmonic Mean (H)

First, calculate the sum of the reciprocals of the numbers:

\(\frac{1}{210} + \frac{1}{252} + \frac{1}{210} = \frac{2}{210} + \frac{1}{252} = \frac{1}{105} + \frac{1}{252}\)

To add these fractions, we find a common denominator. The least common multiple (LCM) of 105 and 252:

• Prime factorization of \(105 = 3 \times 5 \times 7\)
• Prime factorization of \(252 = 2^2 \times 3^2 \times 7\)
• LCM(105, 252) = \(2^2 \times 3^2 \times 5 \times 7 = 4 \times 9 \times 5 \times 7 = 36 \times 35 = 1260\)

Now, rewrite the fractions with the common denominator:

\(\frac{1}{105} = \frac{1 \times 12}{105 \times 12} = \frac{12}{1260}\)

\(\frac{1}{252} = \frac{1 \times 5}{252 \times 5} = \frac{5}{1260}\)

Sum of reciprocals \( = \frac{12}{1260} + \frac{5}{1260} = \frac{17}{1260}\)

Now substitute this into the Harmonic Mean formula:

\(H = \frac{3}{\frac{17}{1260}} = 3 \times \frac{1260}{17} = \frac{3780}{17}\)

Step 4: Calculate the Value of \( \frac{270}{H} \)

Finally, we need to find the value of \( \frac{270}{H} \):

\(\frac{270}{H} = \frac{270}{\frac{3780}{17}}\)

To divide by a fraction, we multiply by its reciprocal:

\(\frac{270}{H} = 270 \times \frac{17}{3780}\)

Simplify the fraction \( \frac{270}{3780} \):

\(\frac{270}{3780} = \frac{27}{378}\)

Divide both numerator and denominator by 9:

\(\frac{27 \div 9}{378 \div 9} = \frac{3}{42}\)

Divide both numerator and denominator by 3:

\(\frac{3 \div 3}{42 \div 3} = \frac{1}{14}\)

So, \( \frac{270}{3780} = \frac{1}{14} \).

Substitute this back into the expression for \( \frac{270}{H} \):

\(\frac{270}{H} = \frac{1}{14} \times 17 = \frac{17}{14}\)

The value of \( \frac{270}{H} \) is \( \frac{17}{14} \).

Revision Table: Harmonic Mean and Binomial Coefficients

Additional Information: Properties of Harmonic Mean and Combinations

The Harmonic Mean is one of the three classic Pythagorean means (Arithmetic Mean, Geometric Mean, and Harmonic Mean). It is typically used when dealing with rates or ratios. For positive numbers, the relationship between these means is AM \( \ge \) GM \( \ge \) HM.

Binomial coefficients, denoted by \(nCr\), represent the number of ways to choose \(r\) elements from a set of \(n\) elements without regard to the order. They are important in combinatorics, probability, and the expansion of binomial expressions like \((x+y)^n\).

The binomial coefficients used in this problem, \(10C4\), \(10C5\), and \(10C6\), are adjacent coefficients in Pascal's triangle. There are interesting relationships between adjacent binomial coefficients, which can sometimes simplify calculations or provide alternative approaches.