Counting question

- mathmath333

Counting question

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- Rushwr

??????? question ?

- mathmath333

\(\large \color{black}{\begin{align}
\normalsize \text{In how many ways can u distribute 7 identical gifts among 5 children.}\hspace{.33em}\\~\\
\end{align}}\)

- mathmath333

ok

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## More answers

- ganeshie8

say the stars are gifts :
\[*~*~*~*~*~*~*\]

- ganeshie8

you want to split those 7 gifts into 5 parts, so place 4 bars in between them :
\[*~*|~*|~*|~*~*|~*\]

- ganeshie8

that arrangement represents :
2 gifts to first child
1 gift to second child
1 gift to third child
2 gifts to fourth child
1 gift to fifth child

- ganeshie8

see if you can tell what below arrangement represents :
\[*|~*~*|~*|~*~*|~*\]

- mathmath333

1 gifts to first child
2 gift to second child
1 gift to third child
2 gifts to fourth child
1 gift to fifth child

- ganeshie8

Perfect!
notice, that string has 7 stars and 4 bars, so the total length of that string is 7+4 = 11

- ganeshie8

as you can see, the problem translates to finding the number of ways of choosing 4 positions for the bars from the 11 positions

- mathmath333

I think now the answer is 11C4. as u are seleecting 11 objects in 4 ways

- ganeshie8

how many ways can you choose 4 different things(positions) from 11 different things(positions) ?

- mathmath333

11C4

- ganeshie8

Thats it!

- anonymous

stars and bars indeed

- ganeshie8

Alternatively you could also think of it as forming different 11 letter words using 7 stars and 4 bars : 11!/(4!*7!)

- mathmath333

\(\large \color{black}{\begin{align}
& \normalsize \text{In how many ways can u distribute 7 identical gifts among 5 children.}\hspace{.33em}\\~\\
& \normalsize \text{such that each child gets at least 1 gift.}\hspace{.33em}\\~\\
\end{align}}\)

- anonymous

reserve 5 gifts, distribute the remaining 2, and then count the ways you can assign one of the reserved 5 to each child

- ganeshie8

you may use the same trick, consider 7 stars :
\[*~*~*~*~*~*~*\]

- ganeshie8

you want to partition that into 5 nonempty parts,
at what positions are you allowed to place the 4 bars ?

- ganeshie8

is below a valid arrangement ?
\[*|~*|~*|~*~*|~*|~*\]

- mathmath333

except the rear and front ends ?

- mathmath333

yes,valid

- ganeshie8

im asking specifically if above arrangement is valid

- ganeshie8

good :)

- ganeshie8

how about below one :
\[|*~*|~*~*~*|~*|~*\]
what does it represent and is it a valid one ?

- mathmath333

0 gifts to first child
2 gift to second child
3 gift to third child
1 gifts to fourth child
1 gift to fifth child
invalid

- ganeshie8

right, that means you don't like the first child
what about below one :
\[*~*|~|~*~*~*|~*|~*\]

- mathmath333

2 gifts to first child
0 gift to second child
3 gift to third child
1 gifts to fourth child
1 gift to fifth child
invalid

- ganeshie8

so you cannot place bars next to each other
and you cannot place bars on the ends

- ganeshie8

the only valid places for bars are :
\[*~-*-~*-*-*-*-*\]
those 6 dashes

- ganeshie8

four bars to place
six positions to choose from
how many total ways can u do it ?

- mathmath333

6C4 ?

- ganeshie8

Yep!

- ganeshie8

a bit more generally,
the number of positive integer solutions to the equation \(\large a+b+c+d+e=n\) is given by \(\large \dbinom{n-1}{4}\)

- ganeshie8

similarly, the number of "non negative" integer solutions to the equation \(\large a+b+c+d+e=n\) is given by \(\large \dbinom{n+4}{4}\)

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