help me loveeeeeeees?

Question

three friends have a total of 6 identical pencils and each one has at least 1 pencil in how many ways can this happen?
a. 1
b. 3
c. 6
d. 10
e. 12

Answer 1

1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
2 2 2
4 1 1
1 4 1
1 1 4

There's the 10 ways!!!

Answer 2

10 ways:

You can divide the pencils among the 3 friends as follows:
1 - 1 - 4
1 - 2 - 3
1 - 3 - 2
1 - 4 - 1
2 - 1 - 3
2 - 2 - 2
2 - 3 - 1
3 - 1 - 2
3 - 2 - 1
4 - 1 - 1

Another way to think of this is to put the 6 pencils on the table.
I ... I ... I ... I ... I ... I

You need to place two dividers between these pencils to separate them into 3 different groups.

You have 5 choices for where to place the first divider.
4 choices for where to place the second divider
But order of the two dividers doesn't matter.

(If I place the first at space 1 and the second at space 2, that's the same as if I placed the first at space 2 and the second at space 1. So the result has to be divided by 2)

So it would be 5 * 4 / 2 = 10 ways

The answer is: d. 10

Answer 3

First you give one pencil each so that each have "at least one". You then have 3 pencils left to distribute.

You can give all 3 to the same person, and there is 3 ways to do this.

Then you can give 2 to one person, and one to another. You have 3 ways to select the person to receive 2 additional pencils, and two choices each for the remaining one, for a total of 6 ways to distribute.

Then you can give each one extra pencil.

So 3 + 6 + 1 = 10.

Answer is D

Answer 4

D. 10

Answer 5

d. 10 ways

Answer 6

I think it's letter C.6