In how many distinct ways can 4 agricultural engineers and 3 agronomists sit in a single row such that no two agronomists sit adjacent to each other? Enter the total number of arrangements ________ (answer in integer).
Answer Key:
1440
Step-by-Step Solution & Explanation
Use the gap method:
1. Arrange the 4 agricultural engineers in a row:
$N_{\text{eng}} = 4! = 24\text{ ways}$
2. This creates 5 available distinct slots (including ends):
$\_ E_1 \_ E_2 \_ E_3 \_ E_4 \_$
3. Choose 3 slots out of 5 for the 3 agronomists and arrange them:
$P(5, 3) = \frac{5!}{(5-3)!} = 5 \times 4 \times 3 = 60\text{ ways}$
4. Total permutations:
$N = 24 \times 60 = 1440\text{ ways}$