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You are given a bucket with apples and oranges. There are two apples, and the rest oranges. Four children uniformly randomly pick a fruit from the bucket. The probability that all four children pick oranges is A and the probability that two of the four children pick the apples is B. After that, you realize at A = B. What's the total amount of fruits (apples + oranges) in the bucket?

My Attempt

  • Total number = ( N )
  • Number of apples = 2
  • Number of oranges = ( N - 2 )

$$ A = \frac{{\binom{{N-2}}{{4}}}}{{\binom{N}{4}}} $$

$$ B = \frac{{\binom{2}{2} \times \binom{{N-2}}{{2}}}}{{\binom{N}{4}}} $$

Given that ( A = B ), we can equate these probabilities:

$$ \frac{{\binom{{N-2}}{{4}}}}{{\binom{N}{4}}} = \frac{{\binom{2}{2} \times \binom{{N-2}}{{2}}}}{{\binom{N}{4}}} $$

Solving, I get that N = 6. But when I use this value to calculate A and B, I realize that they're not equal. Where am I going wrong?

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    $\begingroup$ @BenjaminWang Oh, I see they're equal, thanks. Do you agree with the general approach used to come to the solution. $\endgroup$ Feb 17 at 8:27

2 Answers 2

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$N$ would be $8$, not $6$. $$\begin{align*} &{{N-2} \choose 4} = {{N-2} \choose 2} \\\\ \Rightarrow \;\;\; & N - 2 = 4 + 2 \\\\ \Rightarrow \;\;\; & N = 8 \\\\ \end{align*}$$

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    $\begingroup$ Thank you so much! $\endgroup$ Feb 17 at 17:40
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Below we are solving the given equation: $$ \binom {N - 2}{4} = \binom {2}{2} \cdot \binom {N - 2}{2}. $$ Trivially, $\binom {2}{2} = 1$, so $$ \binom {N - 2}{4} = \binom {N - 2}{2}. $$ By definition, this is saying $$ \frac {\left( N - 2 \right)!}{4! \, \left( N - 6 \right)!} = \frac {\left( N - 2 \right)!}{2! \, \left( N - 4 \right)!}. $$ Cancelling common factors, rearrrangement yields $$ \frac {\left( N - 4 \right)!}{\left( N - 6 \right)!} = \frac {4!}{2!}, $$ whence we derive $$ \left( N - 4 \right) \left( N - 5 \right) = 12. $$ Expand the left-hand side. Rearrange the terms to see $$ {N}^{2} - 9 N + 8 = 0. $$ Factoring should be easy: $$ \left( N - 1 \right) \left( N - 8 \right) = 0. $$ Since $N - 2 \ge 0$, we finally conclude that $N = 8$, as Haris has suggested.

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  • $\begingroup$ Thanks, I appreciate it! I struggled with evaluating the factorials, and this helped a lot :) $\endgroup$ Feb 17 at 17:40

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