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

For each flower type, say there are $n$ number of flowers. We apply star and bars method for each flower type. $n$ flowers of a type will generate $(n+1)$ spaces we just need to place one bar which will separate them into $2$ for the two girls. To do that we need to select a position:

For roses: $\binom{10+1}{1}$
For sunflowers: $\binom{15+1}{1}$
For daffodils: $\binom{15+1}{1}$

Total number of ways distribution can take place $= 11 \times 16 \times 16 = 2816.$

Correct Answer: $D$

Comments

pretty simplified.

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Answer - D
Number Of Ways Roses Can Be Distributed ={(0,10),(1,9),(2,8),…,(10,0)}={(0,10),(1,9),(2,8),…,(10,0)} - 11 Ways11 Ways Similarly, Sunflowers And Daffodils Can Be Distributed In 1616 Ways Each So, Total Number Of Ways =11×16×16=2816.

 

 

answered Oct 4, 2014 by ankitrokdeonsns Loyal (8.7k points) 
edited Jun 7, 2018 by Arjun

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why cant we think as if we have to divide 40(15+15+10) flowers among two girls?

Answer 2

Answer - D
Number of ways roses can be distributed $= \{ (0, 10), (1, 9), (2, 8), \ldots,(10, 0) \}$ - $11 \text{ ways}$
Similarly, sunflowers and daffodils can be distributed in $16$ ways each
So, total number of ways $= 11 \times 16 \times 16 = 2816.$

Answer 3

Combination with repeated objects :

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

Here there are $2$ girls. So, $k =2$

1. $10$ Roses

So $n=10$. Substitute in the formula, we get, $^{10+2-1}C_10=11$                   -------- 1

2. $15$ Sunflowers

So, $n=15$. Substitute in the formula, we get, $^{15+2-1}C_15=16$                ---------- 2

3. $15$ Daffodils

So, $n=15$. Substitute in the formula, we get, $^{15+2-1}C_15=16$                ---------- 3

Total ways = $11*16*16=2816.$

$\therefore$, Option $D$.

Comments

If I compare this with balls-bins problem, then girls are balls & flowers are bins, right ??

I want to say that, how many ways 2 girls can be distributed into 10, 15 & 15 flowers respectively.

This is the question???

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It is the other way around, girls correspond to bins and flowers to balls.

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Thank you buddy for explaining :-)

Answer 4

x1+x2=10 where  x1,x2>=0 so c(2+10-1,1)=11

x1+x2=15 where x1,x2>=0 so c(2+15-1,1)=16

x1+x2=15 where x1,x2>=o so c(2+15-1,1)=16

so 11.16.16=2816