GATE 2023 IN GA Q.3: solution and answer

GATE 2023 · INGA Q.3 · 1 mark · MCQ

A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

(A)
125
(B)
720
(C)
60
(D)
80

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Answer and solution

ANSWER

There are $5 \times 4 \times 4 = 80$ frabjous numbers, so the answer is (D) 80.

Official answer: DSource: Official answer key, GATE 2023 IN (IIT Kanpur)

The odd digits are $1, 3, 5, 7, 9$ , so each position can take one of $5$ digits. Zero is even and can never appear, so there is no leading-zero case to exclude.
Hundreds digit: any odd digit, so $5$ choices.
Tens digit: odd and different from the hundreds digit (the two are adjacent), so $5 - 1 = 4$ choices.
Units digit: odd and different from the tens digit only. It is not adjacent to the hundreds digit, so it may repeat that digit ( $131$ is allowed). That gives $4$ choices.
The choices are made one after another and each count does not depend on which digit was picked earlier, so by the multiplication principle: $5 \times 4 \times 4 = 80$ .
Check with the complement. All-odd 3-digit numbers: $5^{3} = 125$ . Let $P$ = first two digits equal ( $5 \times 5 = 25$ numbers) and $Q$ = last two digits equal ( $25$ numbers). $P \cap Q$ = all three digits equal ( $5$ numbers). By inclusion-exclusion, numbers with at least one equal adjacent pair $= 25 + 25 - 5 = 45$ . So frabjous numbers $= 125 - 45 = 80$ .
The answer is $80$ , which is option (D).

KEY IDEA

Count position by position. The first digit has 5 odd choices. Each later digit must differ only from the digit right before it, so it has 4 choices. The multiplication principle then gives $5 \times 4 \times 4$ .

WATCH OUT FOR

Reading "no two adjacent digits the same" as "all three digits different" gives $5 \times 4 \times 3 = 60$ , option (C). That wrongly rules out numbers like $131$ , where only the non-adjacent first and last digits match. Ignoring the adjacency rule altogether gives $5^{3} = 125$ , option (A).

From the GATE 2023 IN paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes

GATE 2023 IN GA Q.3: solution and answer

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