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How many distinct ways are there to split 50 identical coins among three people so that each person gets at least 5 coins?

1. $3^{35}$
2. $3^{50}-2^{50}$
3. $\begin{pmatrix} 35 \\ 2 \end{pmatrix}$
4. $\begin{pmatrix} 50 \\ 15 \end{pmatrix}. 3^{35}$
5. $\begin{pmatrix} 37 \\ 2 \end{pmatrix}$
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Distinct ways are there to split 50 identical coins among three people so that each person gets at least 5 coins

x1+5+x2+5+x3+5 = 50

x1+x2+x3 = 35

Solving Non integral solution n=35 ,r =3

n+r-1 C r-1 = 35+3-1 C 3-1 = 37 C 2

answered by Veteran (12.5k points) 10 47 90
edited
isn't r=3?
Yes,it was a typo...corrected it
What is the answer if the coins are not distinct?
+1 vote

This is equivalent to say Distribution of  50 non-distinct coins into 3 distinct people where each people gets atleast 5 coins.

First, distribute 5 coins to each people.Then the number of coins left is 35.Then Distribute 35 coins to 3 peoples arbitrarily.

So, total no of ways of Distributing 35 coins to 3 peoples arbitrarily (i.e people get 0 or more coins) = C(3+35-1, 35)  = C(37, 35) =  C(37, 2) = 666

## The correct answer is,(E) C(37, 2)

answered by Boss (8.9k points) 3 8 12
First 5 coins r distributed to everyone .... so it makes 50-15= 35

Now 35 coins will be distributes among 3 peoples arbitrarily ... hence

x1+x2+x3= 35

Using Generating function it will be coefficient of x35

[x^35] [x^0+X^1+....+X^35]^3  [As 35 coins r distributed among 3 people]

=[x^35] [1-x^36]^3 [1-x]^-3

solving it The coefficient will be C(3+35-1,35) =C(37,2)

Hence E is the ans .. Correct me if i am wrong ...
answered by Loyal (3.3k points) 3 7 13