Keys on the table!

Probability Level 3

Assume there are $20$ keys on a table, $10$ are white, and $10$ are black (Every key of the same color are indistinguishable).

(A) How many ways can the 20 keys be arranged? Rotations and reflections are counted.

(B) If $2$ keys are randomly picked, what is the probability of 1 being white and the other black? (If the probability is $\frac{m}{n}$ , (B) would be $m+n$ ).

(D) Is (B) prime? (If answer is yes, (D) would be $3$ , If answer is no, (D) would be $2$ ).

(E) Is (C) prime? (If answer is yes, (E) would be $3$ , If answer is no, (E) would be $2$ ).

And finally:

(F) What is A+B+C+D+E?

The answer is 184791.

1 solution

Micah Gadbois
Jul 18, 2017

(A): $2^{19}$ $=$ $524288$ ( $2$ ways for first key, $2$ ways for second key, $.....)$

(B): There are $20C2$ $=$ $190$ ways to choose $2$ keys. There are $100$ ways to pick $1$ white and $1$ black key. $\frac{100}{190}$ $=$ $\frac{10}{19}$ . Thus, $10+19$ $=$ $29$

(C): (A) is not prime, so answer is $2$

(D): (B) is prime, so answer is $3$

(E): (C) is prime, so answer is $3$

(F): $524288+29+2+3+3$ $=$ $524325$

Edit: This is incorrect. Look at other Solution

In general, avoid making your questions as "take this number and do a bunch of stuff to it". Instead, it would be better to allow people to focus on the crux of your problem directly.

A lot of your answers are wrong.

$A = { 20 \choose 10} = 184756$ ,
$B = \frac{1}{2} \rightarrow 3$ ,
C - no - 2
D - yes - 3
E - yes (always prime) - 3
F - 184767

I have updated the answer accordingly.

Calvin Lin Staff - 3 years, 10 months ago

That's not correct. In part B, the probability to pick any color is 1. There are 10 of the other color, and 19 left, so B is 10/19. 10+19 = 29

Micah Gadbois - 3 years, 10 months ago

Alright, let me run the calculations.

Given such an arrangement, there are $10 \times 9 \times 2$ ways that we can pick 2 ordered keys with different colors. There are $20 \times 19$ ways to pick 2 ordered keys. Hence, the probability is $\frac{ 9}{19}$ .

So this gives us B = 28, D = 2. I have updated the answer to 184791.

Calvin Lin Staff - 3 years, 10 months ago

Keys on the table!

The answer is 184791.

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