How many 4 digit numbers can be formed from the digits 1 to 9 if repetition of digits is not allowed?

Click here to see ALL problems on Permutations

Question 524252: how many 4-digit numbers can be formed with the 10 digits 0,1,2,3...9 of (a)repetitions are allowed,(b)not allowed,(c) the last digit must be 0 and repetitions are not allowed
Answer by Edwin McCravy(19211)

(Show Source):

You can put this solution on YOUR website!
how many 4-digit numbers can be formed with the 10 digits 0,1,2,3...9 of
(a)repetitions are allowed,

Nội dung chính Show

Example 10 - Chapter 7 Class 11 Permutations and Combinations (Term 2)
How many 4 digit number can be formed using the digits 1 to 9 if repetition of digits is not allowed?
How many 4 digit combinations are there using 1 9?
What are the 4 digit combinations 0
How many 4 digit numbers are there if repetition of digit is allowed?

Two ways to get the answer: 1. There are 9999 integers starting with 1 and ending with 9999. But the 999 integers starting with 1 and ending with 999 have less than 4 digits, so the desired number is 9999-999 or 9000 ways. 2. There are 9 ways to pick the first digit (1 through 9), there are 10 ways to pick the 2nd, 3rd, and 4th digits, so that 9�10�10�10 = 9000 ways.
(b)not allowed,
There are 9 ways to pick the first digit (it can't be 0), 9 ways to pick the 2nd digit (it can be 0, just not what we picked for the 1st digit), 8 ways to pick the third digit, and 7 ways to pick the fourth digit. That's 9�9�8�7, or 4536 ways.
(c) the last digit must be 0 and repetitions are not allowed:
We can choose the first digit any of 9 ways, the second digit any of 8 ways, the third digit any of 7 ways and the last digit only 1 way (a 0). That's 9�8�7�1 = 504 ways. Edwin

The digits chosen must sum to a multiple of 3, but not to a multiple of 9. If no repeated digits are allowed, the combinations of digits that have the appropriate sums are
{1, 4, 7, 9}, {2, 4, 6, 9}, {2, 6, 7, 9}
These each can be arranged in 4!=24 ways, to give a total of 3*24 = 72 unique numbers with no repeated digits.

If digits are allowed to be repeated, there are 28 choices. When digits are repeated, the number of possible variations in the digit sequence is reduced. The choices are
{1, 1, 1, 9}, {1, 1, 2, 2}, {1, 1, 4, 6}, {1, 1, 4, 9}, {1, 1, 6, 7}, {1, 2, 2, 7},
{1, 2, 6, 6}, {1, 2, 9, 9}, {1, 4, 4, 6}, {1, 4, 7, 9}, {1, 6, 7, 7}, {1, 7, 7, 9},
{2, 2, 2, 6}, {2, 2, 2, 9}, {2, 2, 4, 4}, {2, 2, 4, 7}, {2, 4, 6, 9}, {2, 4, 9, 9},
{2, 6, 6, 7}, {2, 6, 7, 9}, {4, 4, 4, 9}, {4, 4, 6, 7}, {4, 4, 7, 9}, {4, 6, 7, 7},
{6, 6, 6, 6}, {6, 6, 9, 9}, {6, 9, 9, 9}, {7, 7, 7, 9}

Altogether, there are 295 different numbers that can be made with these sets of digits.

Chapter 7 Class 11 Permutations and Combinations
Serial order wise
Examples

Check sibling questions

Example 10 - Chapter 7 Class 11 Permutations and Combinations (Term 2)

Last updated at May 29, 2018 by

This video is only available for Teachoo black users

Solve all your doubts with Teachoo Black (new monthly pack available now!)

Chapter 7 Class 11 Permutations and Combinations
Serial order wise

Transcript

Example 10(Method 1) How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed? n = Numbers from 1 to 9 = 9 r = 4 Required 4 digit number = 9P4 = 9!/((9 4)!) = 9!/5! = (9 8 7 6 5!)/5! = 3024 Example 10(Method 2) How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed? Let the 4 digit number be Number of 4 digit numbers which can be formed = 9 8 7 6 = 3024

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

How many 4 digit number can be formed using the digits 1 to 9 if repetition of digits is not allowed?

=9×8×7×6=3024.

How many 4 digit combinations are there using 1 9?

There are 10,000 possible combinations that the digits 0-9 can be arranged into to form a four-digit code.

What are the 4 digit combinations 0

Therefore I divide 5040 / 24 = 210. So there are 210 different combinations of four digits chosen from 0-9 where the digits don't repeat.

How many 4 digit numbers are there if repetition of digit is allowed?

Answer. 120 numbers. Case:2 when repetition is allowed. Digits in thousands, hundreds ,tens and unit places can be chosen in 5 different ways respectively.