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Formation of numbers with digits when repetition of digits is not allowed.
First of all decide of how many digits the required numbers will have.
Then fill up the places on which there are restrictions and then apply the formula nPr for filling up the remaining places with remaining digits.
Thousands place 
Hundreds place 
Tens place 
Units place 
Here, Since number must be of four digits and greater than 2300, therefore any one of the five digits 2, 3, 4, 5 and 6 will occur at thousands' place. When any one of 3, 4, 5 and 6 occurs at thousands' place the number will be definitely greater than 2300 but when 2 occurs at thousands' place there will be also restriction on hundreds' place to make the number greater than 2300.
Case I: When 2 occurs at thousands' place:
[Thousands' place can be filled up by 2 in one way and hundreds' place can be filled up by any one of the four digits 3, 4, 5 and 6 in four ways.
Remaining two places can be filled up by remaining five digits in ^{5}P_{2} ways.
∴ Number of numbers formed in this case = 1 x 4 x ^{5}P_{2}
= 4 x
5!
/
(5 − 2)!
= 4 x
5!
/
3!
= 4 x
5 x 4 x 3!
/
3!
= 4 x 5 x 4 = 80
Case II: When any one of 3, 4, 5 and 6 occurs at thousands' place.
Thousands' place can be filled up by any one of the four digits 3, 4, 5 and 6 in four ways and remaining three places can be filled up by remaining six digits in
^{6}P_{3} ways.
∴ Numbers of numbers of formed in this case = 4 x ^{6}P_{3}
= 4 x
6!
/
(6 − 3)!
∴ 4 x
6!
/
3!
= 4 x
6 x 5 x 4 x 3!
/
3!
= 4 x 6 x 5 x 4 = 480
∴ Required number = 80 + 480 = 560

