Count the numbers between 999 and 10000 subject to the condition that

1.	Count the numbers between 999 and 10000 subject to the condition that there are(i) no restriction(ii) no digit is repeated(iii) at least one of the digits is repeated
Answer» (i) We have to find 4 digit numbers The 1000’s place can be filled in 9 ways (excluding zero) and the 100’s, 10’s and unit places respectively can be filled in 10, 10, 10 ways (including zero) So the number of numbers between 999 and 10000 = 9 × 10 × 10 × 10 = 9000 (ii) Since 0 is given as a digit we have to start filling 1000’s place. Now 1000’s place can be filled in 9 ways (excluding 0) Then the 100’s place can be filled in 9 ways (excluding one digit and including 0) 10’s place can be filled in (9 – 1) 8 ways and unit place can be filled in (8 – 1) 7 ways So the number of 4 digit numbers are 9 × 9 × 8 × 7 = 4536 ways (iii) Required number of numbers = 9000 – 4536 = 4464 numbers

Count the numbers between 999 and 10000 subject to the condition that there are(i) no restriction(ii) no digit is repeated(iii) at least one of the digits is repeated

Answer»

(i) We have to find 4 digit numbers

The 1000’s place can be filled in 9 ways (excluding zero) and the 100’s, 10’s and unit places respectively can be filled in 10, 10, 10 ways (including zero)

So the number of numbers between 999 and 10000 = 9 × 10 × 10 × 10 = 9000

(ii) Since 0 is given as a digit we have to start filling 1000’s place.

Now 1000’s place can be filled in 9 ways (excluding 0)

Then the 100’s place can be filled in 9 ways (excluding one digit and including 0)

10’s place can be filled in (9 – 1) 8 ways and unit place can be filled in (8 – 1) 7 ways So the number of 4 digit numbers are 9 × 9 × 8 × 7 = 4536 ways

(iii) Required number of numbers = 9000 – 4536 = 4464 numbers

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