Problem 61 - Algebra and Trigonometry

Question

From a club of 20 people, in how many ways can a group of three members be selected to attend a conference?

Step-by-Step Solution

Answer

The number of ways to select a group of three members from a club of 20 people is 1140.

Step 1: Identify the values of n and r

From the problem, we are told that the club has 20 people (this is our 'n') and we need to select a group of 3 members (this is our 'r'). So, set \(n = 20\) and \(r = 3\).

Step 2: Apply the formula

The formula for combinations is \(C(n, r) = \frac{n!}{r!(n - r)!}\). We substitute the values, getting \(C(20, 3) = \frac{20!}{3!(20 - 3)!}\).

Step 3: Simplify the expressions

Plug in the values and simplify. The expression \(20!\) equals \(20*19*18*17*...*3*2*1\), \(3!\) is \(3*2*1\), and \((20 - 3)!\) or \(17!\) equals \(17*16*15*...*3*2*1\). Use the fact that \(20! = 20*19*18*17!\) to cancel out the equal terms from the numerator and denominator, so that you have \(\frac{20*19*18}{3*2*1}\).

Step 4: Calculate the final answer

After simplifying, we find our answer is \(1140\). This means that it is possible to select a group of three members from a club of 20 in 1140 distinct ways.