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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+n}C_{n }hence we get ^{12}C_{2 }= **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**

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