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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

How many 4-digit numbers abcd are there such that a < b < c < d and...

How many 4-digit numbers abcd are there such that a<b<c<d\mathrm{a} < \mathrm{b} < \mathrm{c} < \mathrm{d} and b−a<c−b<d−c\mathrm{b}-\mathrm{a} < \mathrm{c}-\mathrm{b} < \mathrm{d}-\mathrm{c} ?

Correct answer

07

Step-by-step solution

a<b<c<db−a<c−b<d−c\begin{aligned} & a < b < c < d \\ & b-a < c-b < d-c \end{aligned} possibilities are a=1,b=2,c=4,d=7a=1,b=2,c=4,d=8a=1,b=2,c=4,d=9a=1,b=2,c=5,d=9a=1,b=3,c=6,d= no possible a=2,b=3,c=5d=8a=2,b=3,c=5d=9a=2,b=3,c=6,d= not possible a=3,b=4,c=6,d=9a=4,b=5,c=7,d= not possible \begin{aligned} & a=1, b=2, c=4, d=7 \\ & a=1, b=2, c=4, d=8 \\ & a=1, b=2, c=4, d=9 \\ & a=1, b=2, c=5, d=9 \\ & a=1, b=3, c=6, d=\text { no possible } \\ & a=2, b=3, c=5 d=8 \\ & a=2, b=3, c=5 d=9 \\ & a=2, b=3, c=6, d=\text { not possible } \\ & a=3, b=4, c=6, d=9 \\ & a=4, b=5, c=7, d=\text { not possible } \end{aligned} Ans. 7
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