Question

How many integers greater than 5400 have both of the following properties :
a) the digits are distinct.
b) the digits 2 and 7 do not occur,
​

Answer 1

Answer:

94830.

Step-by-step explanation:

To do this problem we split it into cases. First consider the numbers with 8 digits.

in this case we must order the digits 1, 3, 4, 5, 6, 8, 9, and 0. However when we pick

the digit for the first entry we only have 7 choices since 0 cannot be the first digit.

Then once we have chosen that digit we have to order the remaining 7 digits in any of

the 7! possible permutations. Thus the total number is 7 × 7! = 35280. In a similar

fashion there are 7 × 7! = 35280 seven digit numbers, 7 × 7 × 6 × 5 × 4 × 3 = 17640

six digit numbers of this type and 7 × 7 × 6 × 5 × 4 = 5880 five digit numbers. The

four digit numbers must be handled in cases since they must be bigger then 5400.

First count the numbers that are 6000 or bigger. To do this there are 3 choices for

the first digit (6,8, or 9) then 7 choices for the second, 6 for the third, and 5 for the

fourth position, giving 3 × 7 × 6 × 5 = 630 numbers of this type. Secondly consider

the numbers that begin with a 5. Then there are four choices for the second digit

(4,6, 8, or 9 ) that can be picked for the second digit. Then the third digit must be

chosen from the remaining 6 digits and then there are 5 choices for the fourth digit

Giving 4×6 = 120 numbers of this type. Thus the total number of integers satisfying

conditions (a) and (b) is 120 + 630 + 5880 + 17640 + 35280 + 35280 = 94830.

Answer 2

Answer:

94830

Step-by-step explanation:

How many integers greater than 5400 have both of the following properties :

a) the digits are distinct.

b) the digits 2 and 7 do not occur,

At first go it looks .like that infinite number

but

As digits are distinct

so maximum 10 digits number is possible

but 2 & 7 can not be used so

maximum 8 digit number can be there

Total number =

4 digits numbers

5 Digits numbers

6 digits number

7 digits numbers

8 digits numbers

All the 5 - 8 digit numbers would  be > 5400

Available number of Digits = 8  ( 0 , 1 , 3 , 4 , 5 , 6 , 8 , 9)

for numbers greater than 4 digits (5 - 8 Digits) ,  1st  digit can be  1 , 3 , 4 , 5 , 6 , 8 , 9

then from 2nd digit onward 0 can also be a digit but already used digit can not be used. and further one digit (already used) will keep reducing

5  Digits numbers = 7 * 7 * 6  * 5 * 4  = 5880

6 Digit Numbers =  7 * 7 * 6  * 5 * 4 *3 =  17640

7  Digits numbers = 7 * 7 * 6  * 5 * 4*2  = 35280

8 Digit Numbers =  7 * 7 * 6  * 5 * 4 *3*2*1 =  35280

Now in 4 digit numbers

Number starting with 6 , 8 , 9   > 5400   & number starting with 5

Number starting with 6 , 8 , 9 = 3 * 7 * 6 * 5 = 630

Number starting with 5  , second digit must be 4 , 6  , 8 , 9

Number starting with 5 = 1 * 4* 6 * 5 = 120

Total integers greater than 5400

= 5880 + 17640 + 35280 + 35280 + 630 + 120

= 94830