Let go about it using the slots method.
i. We start by deconstructing the question into it's constituent conditions.
ii. We start with the most restrictive condition first.
So here's our question broken down into it's individual informative components:
~> A seven-digit combination lock on a safe
~> has zero exactly three times,
~> does not have the digit 1 at all.
~> What is the probability that exactly 3 of its digits are odd?
Now let's rearrange those in the order of most restrictive condition down to the least restrictive one:
1. the code has 7 digits
2. has zero exactly three times,
3. does not have the digit 1 at all.
4. exactly 3 of its digits are odd
Not much of a change! Good!
Now let's construct the solution in the same exact order :
1. the code has 7 digits
__ __ __ __ __ __ __
2. has zero exactly three times
0 0 0 __ __ __ __ (the Zeroes can be anywhere, we are only interested in the count, at this stage)
3. does not have the digit 1 at all.
Let's work with only the 4 blank slots from now on.
__ __ __ __ <-- Each of those slots can be filled with any digit among 2, 3, 4, 5, 6, 7, 8, 9 (and at this stage, the probability of choosing any digit for each individual slot is 1/8).
4. exactly 3 of its digits are odd
__ __ __ __ <-- 3 of those slots can be filled with any ODD digit among 2, 3, 4, 5, 6, 7, 8, 9 (Probability = 4/8*4/8*4/8) and the last remaining slot MUST be filled with an EVEN digit among 2, 3, 4, 5, 6, 7, 8, 9 (Probability = 4/8).
Therefore, the code must look something like:
Event 1 : (odd odd odd even) --> Probability = 4/8*4/8*4/8*4/8 = 1/16
or,
Event 2 : (odd odd even odd) --> Probability = 4/8*4/8*4/8*4/8 = 1/16
or,
Event 3 : (odd even odd odd) --> Probability = 4/8*4/8*4/8*4/8 = 1/16
or
Event 4 : (even odd odd odd) --> Probability = 4/8*4/8*4/8*4/8 = 1/16
Therefore total probability = Probability of (Event 1 + Event 2 + Event 3 + Event 4) = 1/16+1/16+1/16+1/16 = 4/16 = 1/4 -> Option D
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Using Formula
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If the probability of a certain individual event is p, then the probability of it occurring k times in n trials/iterations is:
nCk * p^k * (1 - p)^(n - k)
i.e., 4C3 * (1/2)^3 * (1 - 1/2)^1
= 4*1/16
= 1/4 -> Option D
_________________
Best Regards,
Wizard of Oddz
(Always do your best. What you plant now, you will harvest later.)