Permutation and CombinationHard
Question
All the five digits number in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The 97th number in the list does not contain the digit
Options
A.4
B.5
C.7
D.8
Solution
Let required numbers are in form x y z uv
such that x < y < z < u < v
clearly x ≠ 0
Number starting with 1
remaining 4 place can be filled by 8C4 = 70 (from 2, 3, 4, 5, 6, 7, 8, 9)
Number starting with 2
remaining 4 places can be filled by 7C4 = 35
∴ 97th number will be start with 2
now number starting with 23
remaining 3 places can be filled by 6C3 = 20 ways
number starting with 24
Remaining 3 places can be filled by 5C3 = 10 ways
∴ 97Th number will start with 24
Now number starting with 245
Remaining places can be filled by 4C2 = 6 ways
∴ 97th number started with 246 clearly it can not have the digit 5 as x < y < z < u < v
such that x < y < z < u < v
clearly x ≠ 0
Number starting with 1
remaining 4 place can be filled by 8C4 = 70 (from 2, 3, 4, 5, 6, 7, 8, 9)Number starting with 2
remaining 4 places can be filled by 7C4 = 35∴ 97th number will be start with 2
now number starting with 23
remaining 3 places can be filled by 6C3 = 20 waysnumber starting with 24
Remaining 3 places can be filled by 5C3 = 10 ways∴ 97Th number will start with 24
Now number starting with 245
Remaining places can be filled by 4C2 = 6 ways∴ 97th number started with 246 clearly it can not have the digit 5 as x < y < z < u < v
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