Discrete Math How many positive integer less than 1000 consi

Discrete Math

How many positive integer less than 1000 consist of distinct integer from {1,3,5,7,9}

Please explain well (answer is 85)

Solution

Count 1-digit, 2-digit, and 3-digit numbers separately.

1-digit integers: 1,3,5,7,9 - there are 5 integers.

2-digit numbers: Here notice that the one\'s digit can be filled in 5 ways(among the digits 1,3,5,7,9) and the tens digit can be filled in 4 ways(among the remaining digits)

So, by fundamenta principle of counting,we have the total ways as 5*4 = 20 ways

Likewise in the 3-digit numbers: The one\'s digit can be filled in 5 ways, ten\'s digit in 4 ways and the hundredth digit in 3 ways.

So, by fundamenta principle of counting,we have the total ways as 5*4*3 = 60 ways

So, total ways = total numbers = 5+20+60=85