10) (2 points) in a group there are three professors and three doctors. if they are seated in a row, find…

Explanation:

Step1: Calculate total number of arrangements

The total number of people is (3 + 3=6). The total number of ways to arrange (n) distinct objects in a row is (n!). So the total number of arrangements of 6 people in a row is (n = 6), and the total arrangements (N=6! = 720).

Step2: Treat groups of same - profession as single entities

Treat the 3 professors as one unit and the 3 doctors as one unit. Then the number of ways to arrange these two units is (2!). And the number of ways to arrange the 3 professors within their unit is (3!), and the number of ways to arrange the 3 doctors within their unit is also (3!). So the number of favorable arrangements (M = 2!\times3!\times3!) [M=2\times6\times6= 72]

Step3: Calculate the probability

The probability (P) is the number of favorable arrangements divided by the total number of arrangements. So (P=\frac{M}{N}=\frac{72}{720}=\frac{1}{10})

Answer:

(\frac{1}{10})