A secure vault is protected by a unique 3-digit code, where, the code must have no repeated digits, the first digit cannot be zero, and the code contains exactly one digit equal to 7. An intruder knows only this much and tries a random valid code that satisfies all the above conditions. What is the probability that he opens the vault on the first attempt?
(a) 1/200
(b) 1/250
(c) 1/243
(d) 1/300

Question

A secure vault is protected by a unique 3-digit code, where, the code must have no repeated digits, the first digit cannot be zero, and the code contains exactly one digit equal to 7. An intruder knows only this much and tries a random valid code that satisfies all the above conditions. What is the probability that he opens the vault on the first attempt?
(a) 1/200
(b) 1/250
(c) 1/243
(d) 1/300

ONLiNE UPSC · Accepted Answer

Total valid codes with exactly one 7 (and no repeated digits, first digit &ne; 0):
We place 7 in one of the 3 positions, then choose remaining 2 digits from the other 9 (excluding 7), ensuring the first digit &ne; 0.
Let&rsquo;s count all such valid codes:
Case 1: 7 is in the first digit
Choices:
First digit = 7
Remaining 2 digits from 9 (excluding 7), no repetition &rarr; 9P2 = 72
So, 72 codes
Case 2: 7 is in second digit
First digit &ne; 0 or 7 &rarr; choices = 8 (from 1&ndash;9 except 7)
Third digit &ne; first digit or 7 &rarr; choices = 8
So, total = 8 &times; 1 &times; 8 = 64 codes
Case 3: 7 is in third digit (similar to above)
First digit &ne; 0 or 7 &rarr; 8 choices
Second digit &ne; first digit or 7 &rarr; 8 choices
Total = 8 &times; 8 &times; 1 = 64 codes
Total valid codes with exactly one 7 = 72 + 64 + 64 = 200
So, probability = 1/200

Hence, option a is correct.

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M Question 1	⏱ 0
A secure vault is protected by a unique 3-digit code, where, the code must have no repeated digits, the first digit cannot be zero, and the code contains exactly one digit equal to 7. An intruder knows only this much and tries a random valid code that satisfies all the above conditions. What is the probability that he opens the vault on the first attempt? (a) 1/200 (b) 1/250 (c) 1/243 (d) 1/300