Counting question

Question

Counting question

rushwr · Answer

??????? question ?

mathmath333 · Answer

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

mathmath333 · Answer

ok

ganeshie8 · Answer

say the stars are gifts : 

$$*~*~*~*~*~*~*$$

ganeshie8 · Answer

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

ganeshie8 · Answer

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 · Answer

see if you can tell what below arrangement represents :
$$*|~*~*|~*|~*~*|~*$$

mathmath333 · Answer

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 · Answer

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

ganeshie8 · Answer

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 · Answer

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

ganeshie8 · Answer

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

mathmath333 · Answer

11C4

ganeshie8 · Answer

Thats it!

anonymous · Answer

stars and bars indeed

ganeshie8 · Answer

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

mathmath333 · Answer

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

anonymous · Answer

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

ganeshie8 · Answer

you may use the same trick, consider 7 stars : 

$$*~*~*~*~*~*~*$$

ganeshie8 · Answer

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

ganeshie8 · Answer

is below a valid arrangement ?
$$*|~*|~*|~*~*|~*|~*$$

mathmath333 · Answer

except the rear and front ends ?

mathmath333 · Answer

yes,valid

ganeshie8 · Answer

im asking specifically if above arrangement is valid

ganeshie8 · Answer

good :)

ganeshie8 · Answer

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

mathmath333 · Answer

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 · Answer

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

mathmath333 · Answer

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 · Answer

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

ganeshie8 · Answer

the only valid places for bars are : 


$$*~-*-~*-*-*-*-*$$


those 6 dashes

ganeshie8 · Answer

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

mathmath333 · Answer

6C4 ?

ganeshie8 · Answer

Yep!

ganeshie8 · Answer

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 · Answer

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