shashwats wrote:If 5 numbers are to be selected from integers from 1 to 20, inclusive, without replacement, what is the probability that both 5 and 15 will be selected?
(A) 1/57
(B) 1/19
(C) 3/19
(D) 4/57
(E) 2/19
An alternate approach:
Let Y = selecting 5 or 15 and N = selecting any number other than 5 or 15.
ONE WAY to select both 5 and 15: YYNNN
P(Y on the 1st pick) = 2/20. (Of the 20 numbers, either 5 or 15.)
P(Y on the 2nd pick) = 1/19. (Of the 19 remaining numbers, either 5 or 15, whichever was not selected on the first pick.)
P(N on the 3rd pick) = 1. (Since 5 and 15 have already been selected, the 3rd number selected must be a number other than 5 or 15.)
P(N on the 4th pick) = 1. (Since 5 and 15 have already been selected, the 4th number selected must be a number other than 5 or 15.)
P(N on the 5th pick) = 1. (Since 5 and 15 have already been selected, the 5th number selected must be a number other than 5 or 15.)
Since we want all of these outcomes to happen, we multiply the probabilities:
2/20 * 1/19 * 1 * 1 * 1 = 1/10 * 1/19.
TOTAL POSSIBLE WAYS:
YYNNN represents only ONE WAY to select both 5 and 15.
Now we must account for ALL OF THE WAYS to select both 5 and 15.
Any arrangement of the letters YYNNN represents a way to select both 5 and 15.
To illustrate:
NNNYY = selecting 5 and 15 on the 4th and 5th picks.
NYNYN = selecting 5 and 15 on the 2nd and 4th picks.
To account for ALL OF THE WAYS to select both 5 and 15, the product above must be multiplied by the number of ways to arrange YYNNN.
The number of ways to arrange 5 letters = 5!.
But the arrangement here includes IDENTICAL ELEMENTS.
When an arrangement includes identical elements, we must divide by the number of ways to arrange the identical elements.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! (to account for the two identical Y's) and by 3! (to account for the three identical N's).
Thus:
Number of ways to arrange YYNNN = 5!/(2!3!) = 10.
Multiplying the two results, we get:
P(selecting both 5 and 15) = 1/10 * 1/19 * 10 = 1/19.
The correct answer is
B.
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