Question

The total number of four-digit numbers $xyzt$ such that $x<y=z>t$ is

• A. 240
• B. 360
• C. 480
• D. 100

Answer 1

The correct option is A

240

Given: $x<y=z>t$

First, lets fix $y=z=9$, then

8 possibilities(1 to 8) < 9 (fixed) > 9 possibilities(0 to 8), so total number of four-digit numbers $=8\times 9=72$

Similarly, 7 possibilities(1 to 7) < 8 (fixed) > 8 possibilities(0 to 7), so, total number of four-digit numbers $=7\times 8=56$

6 possibilities(1 to 6) < 7 (fixed) > 7 possibilities(0 to 6), so, total number of four-digit numbers $=6\times 7=42$

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Lastly, 1 possibility < 2 (fixed) > 2 possibilities, so, total number of four-digit numbers $=1\times 2=2$

$\Rightarrow$ Total numbers $=72+56+42+30+20+12+6+2=240$