Out of 4 architects, 6 engineers and 9n electricians, how many ways can a group of 6 people be formed to attend a forum if there are at least 3 electricians?
To find the number of ways a group of 6 people can be formed with at least 3 electricians, we can use the concept of combinations.
Step 1: Selecting the electricians
Since we need at least 3 electricians, we can select 3, 4, 5, or 6 electricians.
Case 1: Selecting 3 electricians
There are 9 electricians available, and we need to choose 3 out of them. We can calculate this using the combination formula:
C(9, 3) = 9! / (3! * (9-3)!) = 84 ways
Case 2: Selecting 4 electricians
Similarly, there are 9 electricians available, and we need to choose 4 out of them:
C(9, 4) = 9! / (4! * (9-4)!) = 126 ways
Case 3: Selecting 5 electricians
Again, there are 9 electricians available, and we need to choose 5 out of them:
C(9, 5) = 9! / (5! * (9-5)!) = 126 ways
Case 4: Selecting all 6 electricians
There are 9 electricians available, and we need to choose all 6 of them:
C(9, 6) = 9! / (6! * (9-6)!) = 84 ways
Step 2: Selecting the non-electricians
After selecting the electricians, we need to choose the remaining non-electricians from the remaining pool of architects and engineers. We can select them using the combination formula.
There are 4 architects, and we need to choose 6 - (number of electricians selected) people from them:
C(4, 6 - number of electricians) = 4! / ((6-number of electricians)! * (4 - (6 - number of electricians))!)
There are 6 engineers, and we need to choose 6 - (number of electricians selected) people from them:
C(6, 6 - number of electricians) = 6! / ((6-number of electricians)! * (6 - (6 - number of electricians))!)
Step 3: Calculating the total number of ways
Finally, we need to multiply the number of ways for each case (from Step 1) with the number of ways for selecting the non-electricians (from Step 2) and add them up:
Total number of ways = (C(9, 3) * C(4, 6-3)) + (C(9, 4) * C(4, 6-4)) + (C(9, 5) * C(4, 6-5)) + (C(9, 6) * C(4, 6-6)) +
(C(9, 3) * C(6, 6-3)) + (C(9, 4) * C(6, 6-4)) + (C(9, 5) * C(6, 6-5)) + (C(9, 6) * C(6, 6-6))
Plugging in the values and calculating each term will give us the final answer.