Counting problem

**Hitman6267** · July 11th 2010, 03:47 AM

A women wants to distribute 12 identical cookies to her 5 children of different ages.
The youngest has to get at least 2 cookies while the rest at least 1.
In how many way can she do it ?
The answer is 210

My unsuccessful attempt was dividing the problem
first, we give 1 of the children 2 cookies and the rest 1 cookie.
She can do that in 1 way.

Then we are left with 12-6 = 6 cookies to distribute.
We calculate "6 choose 5".

Can some one help ?

**Plato** · July 11th 2010, 03:58 AM

Originally Posted by Hitman6267

A women wants to distribute 12 identical cookies to her 5 children of different ages.
The youngest has to get at least 2 cookies while the rest at least 1.
In how many way can she do it ?
The answer is 210

There are $\dbinom{N+K-1}{N}$ ways to distribute $N$ identical objects to $K$ different cells.
Let’s start by giving the youngest child two cookies.
Then give each of the other four children one cookie each.
That leaves six cookies to distribute any way the mother wants: $\binom{6+5-1}{6}$ .

**Also sprach Zarathustra** · July 11th 2010, 04:00 AM

Solve the next linear equation:

x_1 +x_2+x_3+x_4+x_5=12

with following conditions:

x_1>=2
x_2,x_3,x_4,x_5>=1

**Soroban** · July 11th 2010, 06:47 AM

Hello, Hitman6267!

A women wants to distribute 12 identical cookies to her 5 children of different ages.
The youngest has to get at least 2 cookies while the rest get at least 1.
In how many way can she do it?
(The answer is 210)

Place the 12 cookies in a row, leaving spaces between them.

. . $\circ\:\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\ ;\_\;\circ\;\_\;\circ \;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\; \;\circ$

Distribute 4 "dividers" among the 11 spaces.
. . There are: . $_{11}C_4 \:=\:{11\choose4} \:=\:330$ ways.

So that: . $\circ\,\circ\,|\,\circ\,\circ\,\circ\,|\,\circ\, \circ\,|\,\circ\,|\,\circ\,\circ\,\circ\,\circ$
. . represents . $\{2,3,2,1,4\}$ from the youngest to the oldest.

And that: . $\circ\,|\,\circ\,\circ\,|\,\circ\,\circ\, \circ\,|\,\circ\,\circ\,\circ\,\circ\,\circ\,|\,\c irc$
. . represents $\{1,2,3,5,1\}.$

But this includes distributions in which the youngest gets only one cookie.

Very well, how many ways are there in which the youngest gets one cookie?

We have: . $\circ\,|\,\circ\,\_\,\circ\,\_\,\circ\,\_\, \circ\,\_\,\circ\,\_\,\circ\,\_\,\circ\,\_\,\circ\ ,\_\,\circ\,\_\,\circ\,\_\,\:\circ$

And we must distribute 3 dividers among the 10 spaces.
. . There are: . $_{10}C_3 \:=\:{10\choose3} \:=\:120$ ways.

Therefore, there are: . $330 - 120 \;=\;\boxed{210}$ ways in which the youngest
. . gets at least 2 cookies and the others get at least one.

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