How many different ways can 14 pencils be passed out to 6 different people? Some people are allowed no pencils.

There are 2 questions that are very similar and I have the same answer to both but I don't think that's correct. Can you help me see the difference between the 2 questions.

1. We have 14 indistinguishable pencils and we want to hand out all of the pencils to 6 people (but we do not care if some of the people get no pencils.) How many different ways could this be done?

My answer: 19 choose 14

2. We have 14 indistinguishable pencils and we want to hand out all of the pencils to 6 people and we want everyone to get at least one pencil. How many different ways could this be done?

My answer: again 19 choose 14

Am I right or wrong or both? I'm confident one answer is 19 choose 14 but I'm not sure which. As you can see I'm a bit confused.

• The first is correct. Can you give reasons for this answer? - if so I think you will see why the second one is wrong, and how to fix it. – David May 1 '15 at 2:13
• By using 5 dividers for the 6 people (think of the 6 people as "types"), putting the 14 pencils in the first section or divider and then counting the remaining sections/dividers which would be 5, so 14+5=19 – Seumas Frew May 1 '15 at 2:14
• @David Since each person would have 1 pencil, would it essentially be like handing out 8 pencils to the 6 people? If so, that would be 13 choose 8. – Seumas Frew May 1 '15 at 2:16
• Yes that's correct. – David May 1 '15 at 2:17

Problem 1 is correct with the answer being $\binom{19}{14}$. Problem 2 is wrong and the correct answer is $\binom{13}{8}$. You reserve 1 pencil to each of the 6 people leaving you with 8 pencils left to pass out. The problem can be looked at as being how many different ways can we pass out 8 pencils to 6 people, and that would be $\binom{13}{8}$.
• @TravisJ Nevermind, I see what you did. $\binom {19}{14}$ – Seumas Frew May 1 '15 at 2:59
I think the second question is easier and should be asked first. Put the pencils in a row and there are $13$ spaces between them. You need to select $5$ of these spaces to stop giving pencils to a given person and start giving them to the next, so there are ${13 \choose 5}={13 \choose 8}$ ways to do this.
The first then comes from adding $1$ to the number given to each person. You now have $20$ pencils to distribute to $6$ people, each of whom must get at least one. The same logic says the answer is $19 \choose 5$, which happens to equal $19 \choose 14$
• The first question results in an equation of the form $x_1 + x_2 + \cdots + x_k = n$ with solutions in the nonnegative integers. The number of solutions is the number of ways we can insert $k - 1$ addition symbols in a row of $n$ ones, which is $\binom{n + k - 1}{k - 1} = \binom{n + k - 1}{n}$. The second equation results in an equation of the same form with solutions in the positive integers. The number of solutions is the number of ways we can fill $k - 1$ of the $n - 1$ spaces between the $n$ ones with addition signs, which is $\binom{n - 1}{k - 1}$, as you clearly recognized. – N. F. Taussig May 1 '15 at 8:30