From his 9 friends (4 boys and 5 girls), Mohan wanted to invite friends for picnic. If there have to be exactly 3 girls in invites, then the number of ways in which he can invite them are,a)320b)80c)160d)200Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To determine the number of ways Mohan can invite 3 girls from his 9 friends (4 boys and 5 girls), we can use the concept of combinations.

Combination formula:
The number of ways to choose r items from a set of n items is given by the combination formula:
C(n, r) = n! / (r! * (n-r)!)

Where n! denotes the factorial of n, which is the product of all positive integers less than or equal to n.

Number of ways to invite 3 girls:
In this case, Mohan needs to invite exactly 3 girls. Therefore, r = 3.
Since there are 5 girls in total, n = 5.

Plugging in these values into the combination formula, we get:
C(5, 3) = 5! / (3! * (5-3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2 * 1)
= (5 * 4) / (2 * 1)
= 10

So there are 10 different ways to invite 3 girls from the group.

Number of ways to invite the remaining friends:
After selecting the 3 girls, Mohan needs to invite the remaining friends. Since there are 4 boys and 2 remaining spots, he can choose any combination of boys.

The number of ways to select 2 boys from a group of 4 is given by:
C(4, 2) = 4! / (2! * (4-2)!)
= 4! / (2! * 2!)
= (4 * 3 * 2!) / (2! * 2 * 1)
= (4 * 3) / (2 * 1)
= 6

Total number of ways to invite friends:
To calculate the total number of ways to invite friends, we multiply the number of ways to choose 3 girls by the number of ways to choose 2 boys:
Total ways = 10 * 6 = 60

Therefore, the number of ways Mohan can invite friends for the picnic with exactly 3 girls is 60.

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