Involving the selection of digits from a set... Select digits from the set {2,4,6,7,8,9} to form a number between 20 and 900...

Question

Here's the whole problem and what it's asking:

Assume that you select digits from the set {2,4,6,7,8,9} to form a number between 20 and 900.

1.How many such numbers can be formed if digits are not repeated?
2.How many such numbers can be formed if digits can be repeated?

 

I started by treating it as a two part problem using the multiplication principle: the first part by finding the number of two digit numbers and the second part by finding the number of 3 digit numbers but I am not getting the correct answer. It's the very last problem I have and it's due at midnight. Any help would be appreciated, I am so lost.

Answer 1

Hi Alexis.

First determine how many numbers exist in the range:

Answer = 899 - 20 = 879 (900 and 20 are not available).

 

First lets work on the 2 digit numbers:

From the set you have for the first and second digit of the number.

Non-Repeat = 6, 5

Repeating = 6, 6

 

Total Combinations for 21-99

Non-Repeat = 30
Repeating = 36

 

Now for the 100 - 899 range

Digit counts assignments: (here the 9 can't be used as the first digit)

Non-Repeat = 5, 5, 4
Repeating = 5,6,6

Total Combinations for the 100 - 899 range
Non-Repeat = 100
Repeating = 180

 

Total combination for 21-899
Non-Repeat = 130
Repeating = 216

 

Answer 2

Hi Alexis,

 

(a) Since the set {2,4,6,7,8,9} has 6 elements, we can use combination

For two digits  number,  we can select the number in ⁶C₂  = 6!/2!4! =15 

For three digits number, we can select  from the remaining ⁴C₃ = 4

There are 15*4 =60 numbers

(b) With repeated digits,

Total numbers = ⁶C₂ * ⁶C₃ =15*20=300