Question

Between 20000 and 70000 find the number of even integers in which no digit is repeated

Answer 1

Given : even integers Between 20000 and 70000

To find:  Number of integers

Solution:

Between 20000 and 70000

Number of even integers  

Starting with 20000

Even number ends up with 0 , 2 , 4 , 6 , 8

1st digit   - 2  

5th digit 0 , 4 , 6 , 8

Case 1 :  5th digit 0

in between 3 digits has to be 1 , 3 , 4 , 5 , 6 , 7 , 8 , 9

3 Digits out of 8 can be selected in ⁸P₃  = 336 Ways

Similar  336 ways for  5th digit ending with 4 , 6 , 8

Hence  336 * 4 = 1344  Ways

1st digit   - 3

5th digit 0 , 2,  4 , 6 , 8

so Possible numbers  =  336 * 5  =  1680

1st digit   - 4

5th digit 0 , 2,  6 , 8

so Possible numbers  =  336 * 4  =  1344

1st digit   - 5

5th digit 0 , 2,  4, 6 , 8

so Possible numbers  =  336 * 5  =  1680

1st digit   - 6

5th digit 0 , 2,  4 , 8

so Possible numbers  =  336 * 4  =  1344

Total Digit = 1344 * 3  + 1680 * 2

= 7392

Bit it includes 20000 also

so subtract 1

Hence 7391  are number of even integers in which no digit is repeated

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Answer 2

Answer:

The number of even integers in which no digit is repeated is 7392.

Step-by-step explanation/solution:

Let us represent this number as ABCDE.

Excluding 70000 (which is not a solution due to given limits):

• A will be one of {2, 3, 4, 5, 6}
• E will be one of {0, 2, 4, 6, 8}

Consider the two-digit number AE:

• If digits were independent (repetition doesn't matter), there would be 5 x 5 = 25 different possibilities for AE.
• However, digits cannot be repeated, so we must exclude the following numbers:  {22, 44, 66}.
• Hence this gives 25 - 3 = 22 cases for AE.

We now apply this logic to the number ABCDE.

There are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

For number B, there are 8 possible digits; 10 digits - 2 digits in A, E.

For number C, there are 7 possible digits; 10 digits - 3 digits in A, B, E.

For number D, there are 6 possible digits; 10 digits - 4 digits in A, B, C, E.

Therefore, using the Multiplication Principle, we find that there are:

22 x 8 x 7 x 6 = 7392 even integers between 20000 and 7000 where no digits are repeated.