In order to conduct an experiment, 4 subjects are randomly selected from a group of 42 subjocts. How many difforent groups of 4 subjects are possible? The number of possible different groups is \( \square . ~ \) (Type a whole number.)

To determine the number of different groups of 4 subjects that can be selected from a total of 42 subjects, we use the combination formula, which is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] Where: - \( n \) is the total number of subjects (42 in this case), - \( r \) is the number of subjects to choose (4 in this case), - \( ! \) denotes factorial, which is the product of all positive integers up to that number. Using this formula, we have: \[ C(42, 4) = \frac{42!}{4!(42-4)!} = \frac{42!}{4! \cdot 38!} \] Calculating this step-by-step: 1. Calculate \( 42! \) divided by \( 38! \): \[ \frac{42 \times 41 \times 40 \times 39 \times 38!}{38!} = 42 \times 41 \times 40 \times 39 \] 2. Calculate \( 4! \): \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] 3. Now substitute back into the combination formula: \[ C(42, 4) = \frac{42 \times 41 \times 40 \times 39}{24} \] 4. Now calculate \( 42 \times 41 \times 40 \times 39 \): \[ 42 \times 41 = 1722 \\ 1722 \times 40 = 68880 \\ 68880 \times 39 = 2688240 \] 5. Finally divide by \( 24 \): \[ \frac{2688240}{24} = 112850 \] Thus, the number of possible different groups of 4 subjects is \( 112850 \).