CSAT Solved Papers/ 2022/Q79

2022 CSAT — Q79

Quant Counting & combinatorics 2.5 marks Easy

What is the number of numbers of the form 0⋅XY0 \cdot XY, where XX and YY are distinct non-zero digits?

  1. A 72 Answer
  2. B 81
  3. C 90
  4. D 100

Worked rationale

XX is a non-zero digit: 99 choices (11–99). YY is a non-zero digit distinct from XX: 88 choices. By the multiplication principle:

9×8=72.9 \times 8 = 72.

Answer: (a) 72.

Why the other options miss

  • B
    dropped “distinct”: uses 9×99 \times 9, allowing X=YX = Y.
  • C
    let YY be zero: lets YY range over 00–99 (9×109 \times 10), ignoring “non-zero” on YY.
  • D
    ignored both rules: counts 10×1010 \times 10, dropping “non-zero” and “distinct” together.

Specialist insight

Two filters tighten the count: both digits non-zero (99 each, not 1010) and distinct (so the second slot loses one option). 9×8=729 \times 8 = 72. Each distractor relaxes exactly one constraint — 8181 drops distinct, 9090 drops one non-zero, 100100 drops both. Encode every word of the constraint into the slot counts.

The trap, in one line

Non-zero and distinct: 9×8=729 \times 8 = 72 (not 9×99\times 9 or 10×1010\times 10) ⇒\Rightarrow (a).

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