Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} C_{r}\) using the formula from this section. $$_{4} C_{1}$$

Step-by-Step Solution

Answer

The value of \(_{4} C_{1}\) is 4.

Step 1: Understand the Formula

\(_{n} C_{r} = \frac{n!}{r!(n-r)!}\) is known as the combination formula. It helps in finding the number of ways selecting r items from a group of n. Here, '!' is a factorial symbol. For any positive integer n, n factorial is the product of all positive integers less than or equal to n. Example: 4! = 4 x 3 x 2 x 1 = 24.

Step 2: Identify Values

In the expression \(_{4} C_{1}\), we have n = 4 and r = 1. We need to substitute these values in place of n and r in the formula.

Step 3: Calculate Factorials

Calculate factorial of the numbers. 4! = 4 x 3 x 2 x 1 = 24 and 1! = 1

Step 4: Substitute Values and Calculate

Substitute the values of r and n into our formula to find the value for \(_{4} C_{1}\). This means we have: \(_{4} C_{1} = \frac{4!}{1!(4 - 1)!} = \frac{24}{1 * (3!) } = \frac{24}{6} = 4\)