I’ll give it a try....

I believe the question is saying “at least one toy and at least one marble”


(1st) distributing the 4 different toys to 3 different people

Each person must receive at least 1 of the 4 toys. This means one person will receive 2 toys and the other two people will receive just 1 toy.


[2 ; 1 ; 1]

Step 1: distribute the different toys into identical stacks as listed above

“4 choose 2” * “2 choose 1” * 1 choose 1” = 4! / (2! * 1! * 1!)

However, since there are identical amounts of toys placed in the stacks that contain 1, we will have repeat counts. To remove the overcounting, we need to divide by 2!

End up with: 4! / (2! * 2!*) = 6 ways

AND

Step 2: Because we have 3 distinct groups of people, we need to arrange the “stacks” among the 3 people. Because each stack has different items, it matters which stack with 1 toy goes to which person.

We can arrange the 3 different stacks among 3 different people in 3! Ways

Toys can be distributed in: 6 * 3! = 36 ways

And

(2nd) distribute 5 identical marbles among 3 distinct “groups” (people)

Letting the 3 different people be variables A B and C

A + B + C = 5

If each person must receive 1, then we are left with the linear equation:

a + b + c = 2

No of ways to distribute the identical marbles = 4! / (2! * 2!) = 6 ways


(36 ways to distribute the 4 toys) * (6 ways to distribute the 5 identical marbles) = 216 ways

D 216

Hopefully I didn’t miss anything (and read the question correctly)

Posted from my mobile device

The 4 toys can be distributed in 2 , 1, 1 ways where each gets at least 1 toy.

The number of ways of doing this is 4C2 * 2C1 * 1C1 = (4 * 3/2) * 2 * 1 = 12 ways

Now the number of ways where 2,1,1 can be rearranged = 3!/2! = 3 ways (211) (121)(112)

Total ways of distributing the toys = 12 * 3 = 36 ways


For the marbles, since they are the same, let us give away 1 marble to each. We are left with 2 marbles. The number of ways of distributing them is

2, 0, 0 = 3!/2! = 3 ways and

1, 1, 0 = 3!/2! = 3 ways

The total ways = 3 + 3 = 6


Therefore the total distribution can be done in 36 * 6 = 216 ways


Option D

Arun Kumar
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