Counting : Beginning Advanced Level

Problems:

# 1: How many 3 digit numbers are there? 2 digit even numbers are there?  

Solution I: a. 3 digit starts with 100 and ends in 999 so there are 999 -100 + 1 = 999 - 99 = 900 three digit numbers
                b.  2 digit numbers start with 10 and ends in 99 so there are 99 - 9 = 90 two digit numbers. 

Solution II: a. ___ ___ ___ You can't have "0" on the hundredth digit so there are 9 numbers (1 to 9) for the hundredth digit, 10 (you can use "0" now) for both the tenth and unit digit.
There are 9 * 10 * 10 = 900 three digit numbers.
                 b. ___   ___ Again, you can't have "0" on the tenth digit so there are 9 * 10 = 90 two digit numbers.

Quite a few students are confused with questions such as this.
Use easier example to demonstrate why you multiply, not add the numbers.

How many ways to pair A, B, C  with 1, 2, 3.
You have (A, 1), (A, 2), (A, 3), (B, 1), (B, 2), (B, 3), (C, 1), (C, 2) and finally (C, 3), which is
3 * 3 = 9 ways to pair them instead of adding 3 + 3.

# 2: How many 3 digit numbers are there that the hundredth digit is larger than 4, the tenth digit is odd and

 the ones (or unit digit) is even? 

Solution:  ___ ___ ___  There are 5 possible numbers for the hundredth digit (5, 6, 7, 8, 9) that is larger than 4,  5 digits that are odd (1, 3, 5, 7, 9) and 5 digits that are even (0, 2, 4, 6, 8) so the answer is 

5 * 5 * 5 = 125 three digit numbers

# 3: How many 3-digit numbers chosen from 2, 4, 5, 7, 9 can you make if you can repeat the digits, but
the same number can't be right next to each other? 

Solution: ___ ___ ___  There are 5 possible digits for the first slot; however, since you can't use the same digit right next to each other, only 4 digits left for the second slot. Again, you can't use the same digit as the

second slot so 4 left for the third slot. 

5 * 4 * 4 = 80  three digit numbers. 

Applicable problems:  Answer key below.

#1: How many three digit positive odd integers?
#2: How many 3 digit odd numbers can be formed with the digits 1, 2, 3, 4, 5 if you can repeat digits?
What about if you can't repeat digits?
#3: How many 3 digit numbers can be formed with the digits 0,1,2, 3, 4, 5 if repeating digits are allowed? How many are even and how many are odd?

#4: How many 3 digit numbers can be formed with the digits 0,1,2, 3, 4, 5 if no digit is repeated in any number? How many are even and how many are odd? (This one is tricky!!)















Answer key:
#1: 9 * 10 * 5 = 450
#2: 5 * 5 * 3 = 75 (digits can be repeated) ; 4 * 3 * 3 = 36 (digits can not repeated)
#3: 5 * 6 * 6 = 180; There will be 5 * 6 * 3 = 90 even numbers and the same 5 * 6 * 3 = 90 odd numbers
Don't forget that "0" can't be placed on the hundredth digit.
#4: 100 numbers can be formed. 48 odd and 52 even numbers.