GATE CSE 1999 | Question: 2.2

Question

Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves?

• A. $1638$
• B. $2100$
• C. $2640$
• D. None of the above

Comments

@Deepak Poonia Okay Sir , I got it.

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doubt in understanding the language of question: 

“divide the flowers among themselves?”

here, why can’t we apply start-bars on the total number of flowers here?
@Deepak Poonia

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“Star-Bar” template is used for “distributing identical objects into distinct boxes”.

 All the flowers collectively are Not identical. If All 40 flowers were identical, the answer would have been $41C1 = 41.$

Answer 1

First let me give you naive solution and then the proper approach or short cut

There are 10 roses,15 sunflowers , 15 daffodils

Roses	Sunflowers	Daffodils
0 – 10	15 – 0	15  – 0
1 – 9	14 – 1	14  – 1
2 – 8	13 –  2	13  – 2
3 – 7	12 – 3	12  – 3
4 – 6	11 – 4	11  – 4
5 – 5	10 – 5	10  – 5
6 – 4	9 – 6	9  – 6
7 – 3	8 – 7	8  – 7
8 – 2	7 – 8	7  – 8
9 – 1	6 – 9	6  – 9
10 -0	5 – 10	5  – 10
	4 – 11	4  – 11
	3 – 12	3  – 12
	2 – 13	2  – 13
	1 – 14	1  – 14
	0 – 15	0  – 15

Number ways of distributing Roses  = 11

Number ways of distributing Sunflower  = 16

Number ways of distributing Daffodils  = 16

Total number of ways of distributing flowers = 11*16*16 =  2816

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n identical objects  can be distributed among kk distinct people in  =$^{n+k−1}$$C_n$ ways

 

Here there are 2 girls. So, k=2

1. 10Roses

So n=10 Substitute in the formula, we get,=$^{10+2−1}$$C_{10}$=11                

2. 15 Sunflowers

So, n=15. Substitute in the formula, we get, $^{15+2−1}$$C_{15}$=16   

3. 15 Daffodils

So, n=15. Substitute in the formula, we get, $^{15+2−1}$$C_{15}$=16          

Total ways = 11∗16∗16=2816.

 

Answer 2

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Answer 3

A solution using generating functions. 

 

​​​​​​Note the number of roses is 10. Hence I will be interested in finding the x^10 coefficient. And for both daffodils and sunflowers we are interested how 15 of them each is divided hence x^15 coefficient.

 

​​​​​G(x) = 1+x+x²+x³+.… = 1/(1-x)

the reason is simple because a girl could get 0flowers, or 1flower or 2flowers and so on. 

suppose if the question was, the girls could get only odd number of flowers, then G(x) would have been, 

x+x³+x⁵+x⁷+… for each girl. 

G(x) = x/(1-x).

If you want to know more read the below given pdf or watch Kimberly Brehm videos on generating functions on YouTube.

Videos on generating functions.

 

 

 

 

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