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Question
If numbers are formed using digits 2, 3, 4, 5, 6 without repetition, how many of them will exceed 400?
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Solution
Digits are 2, 3, 4, 5, 6.
We have to form the numbers greater than 400.
The repetition of digits is not allowed. The numbers greater than 400 may be of
(i) 3-digit numbers:
For 3-digit numbers greater than 400, hundred's place can be filled either by 4 or 5 or 6.
Hundred's place can be filled by using 4 or 5 or 6 in 3 different ways.
The ten's and unit's place can be filled by remaining digits in 4 and 3 ways respectively.
∴ total number of 3-digit numbers greater than 400
= 3 × 4 × 3
= 36
(ii) 4-digit numbers:
The thousand's place can be filled by anyone of the given 5 digits in 5 different ways.
Since repetition of digits is not allowed, the hundred's place, ten's place, and unit's place can be filled by remaining digits in 4, 3, and 2 ways respectively.
∴ total number of 4-digit numbers
= 5 × 4 × 3 × 2
= 120
(iii) 5-digit numbers:
The ten thousand's place can be filled by anyone of the given 5 digits in 5 different ways.
Since repetition of digits is not allowed, the thousand's place, hundred's place, ten's place, and unit's place can be filled by remaining digits in 4, 3, 2, and 1 way respectively.
∴ total number of 5-digit numbers
= 5 × 4 × 3 × 2 × 1
= 120
Thus, the total numbers greater than 400
= 36 + 120 + 120
= 276
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