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

  1. $1638$
  2. $2100$
  3. $2640$
  4. None of the above
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7 Answers

Best answer
58 votes
58 votes
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: $D$
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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 $= 11 \times 16 \times 16 = 2816.$
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13 votes
13 votes

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 $D$.

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1 votes
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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
Answer:

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