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44 = 256 But, if you want to actually exclude those numbers which happen not to include repeated digits (e.g. 4321), then 44 - 4! = 232

Q: How many 4-digit numbers can be formed by the digits 1 2 3 4 using the digits more than once in each number example 1112 1122?

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Numbers formed should be of at least 3 digits means they may be of 3 digits, 4 digits, 5 digits or 6 digits. There are 6 choices for digit in the units place. There are 5 and 4 choices for digits in ten and hundred’s place respectively. So, total number of ways by which 3 digits numbers can be formed = 6.5.4 = 120 Similarly, the total no.of ways by which 4 digits numbers can be formed = 6.5.4.3 = 360. the total no. of ways by which 5 digits numbers can be formed = 6.5.4.3.2 = 720. The total no.of ways by which 4 digits numbers can be formed = 6.5.4.3.2.1 = 720. So, total no.of ways by which the numbers of at least 3 digits can be formed = 120 + 360 + 720 + 720 = 1920.

99999

There are 5 numbers of 1 digit, 25 numbers of 2 digits, and 75 numbers of 3 digits. This makes 105 numbers in all.

There are 90 of them that can be formed by concatenation (writing them side-by-side). You can, of course, form lots more by using mathematical operations. Foe example, 99 = 387,420.489. Combining factorials and powers will give numbers with more than 2 million digits!

500

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Numbers formed should be of at least 3 digits means they may be of 3 digits, 4 digits, 5 digits or 6 digits. There are 6 choices for digit in the units place. There are 5 and 4 choices for digits in ten and hundred’s place respectively. So, total number of ways by which 3 digits numbers can be formed = 6.5.4 = 120 Similarly, the total no.of ways by which 4 digits numbers can be formed = 6.5.4.3 = 360. the total no. of ways by which 5 digits numbers can be formed = 6.5.4.3.2 = 720. The total no.of ways by which 4 digits numbers can be formed = 6.5.4.3.2.1 = 720. So, total no.of ways by which the numbers of at least 3 digits can be formed = 120 + 360 + 720 + 720 = 1920.

There are 7,290 different 4-digit numbers that can be formed from the digits 1-9 without repetition.

99999

There are 5*4*3 = 60 such numbers.

There are 5 numbers of 1 digit, 25 numbers of 2 digits, and 75 numbers of 3 digits. This makes 105 numbers in all.

There are 90 of them that can be formed by concatenation (writing them side-by-side). You can, of course, form lots more by using mathematical operations. Foe example, 99 = 387,420.489. Combining factorials and powers will give numbers with more than 2 million digits!

256

24

500

There are 60480 numbers.

840

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