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How many ways $10$ roses can be distributed among $3$ girls?

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Answer is 66(when at most 10 roses can be given to a girl)

this is stars and bars question,

if we have 10 roses kept linearly on a table then we will need 2 bars to divide roses in 3 parts.

n (roses) = 10

m(bars) = 2

Use the formula m+nCn hence we get 12C2 66

Answer is 36 (when at least 1 rose is distributed among all girls)

Number of ways in which n identical things can be divided into r groups, if blank groups are not allowed (here groups are numbered, i.e., distinct)

= Number of ways in which n identical things can be distributed among r persons, each one of them can receive 1,2 or more items

(n-1)C(r-1)
Apart from this formula you can think logically that if 1st girl gets only one rose then other two can get a sum of 9 flowers which can be

 Girl 1 Girl 2 Girl 3 1 1 8 1 2 7 1 3 6 1 4 5 1 5 4 1 6 3 1 7 2 1 8 1

and if 1st girl gets 2 roses then number of combination decreases by 1

 Girl 1 Girl 2 Girl 3 2 1 7 2 2 6 2 3 5 2 4 4 2 5 3 2 6 2 2 7 1

if 1st girl get 3 roses again the combination will go down by one

 Girl 1 Girl 2 Girl 3 3 1 6 3 2 5 3 3 4 3 4 3 3 5 2 3 6 1

We can see the combinations are going down as the 1st girl is getting a flower more than the prev one so

total combination with 1st girl having 1 rose = 8

total combination with 1st girl having 2 rose = 7

total combination with 1st girl having 3 rose = 6

total combination with 1st girl having 4 rose = 5

total combination with 1st girl having 5 rose = 4

total combination with 1st girl having 6 rose = 3

total combination with 1st girl having 7 rose = 2

total combination with 1st girl having 8 rose = 1

any girl can have maximum of only 8 roses because other girls must get atleast 1

hence total combinations  = 8+7+6+5+4+3+2+1 = 36

by (256 points) 3 6 14
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ans 66
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55?(n+r-1)cr.  n-12.  R-2
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yes 66