Brain Teasers
Less than 3 Million
How many 7 digit EVEN numbers less than 3,000,000 can be formed using the digits, 1, 2, 2, 3, 5, 5, and 6?
Hint
You must account for the identical digits to make sure you don't double count.Answer
If the digit:1 is first and 2 is last, there are 60 ways to form a 7 digit even number less than 3,000,000.
1 is first and 6 is last, there are 30 ways to form a 7 digit even number less than 3,000,000.
2 is first and 6 is last, there are 60 ways to form a 7 digit even number less than 3,000,000.
2 is first and 2 is last, there are 60 ways to form a 7 digit even number less than 3,000,000.
So, adding up all the different combinations that can be formed we get:
60+30+60+60 = 210.
In total, 210 ways to form a 7 digit even number less than 3,000,000 using the digits 1, 2, 2, 3, 5, 5, and 6.
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Comments
um, wow. I'll take your word for it.
It can be reduced to 2 separate cases. Case 1 ends with a 2 and therefore 2 ways to start (120 ways) and case 2 ends with a 6 and therefore 3 ways tostart (90 ways). Good puzzle!
I did it basically as the given answer:
Starts with 1 ends with 2:
C(5,2)*3! = 10*6 = 60
Starts with 1 ends with 6:
C(5,2)*C(3,2) = 10*3 = 30
Starts with 2 ends with 2 or 6:
2*C(5,2)*3! = 2*10*6 = 120
120+60+30 = 210
The notation C(5,2) reads a "5 choose 2" and is the number of combinations when taking two items from a collection of five. It is calculated as
5!/(2!*(5-2)!) =
(5*4*3*2*1)/((2*1)*(3*2*1)) =
(5*4)/(2*1) = 20/2 = 10
In the above calculations, the five things are the internal digit places and the two things are the two fives.
Starts with 1 ends with 2:
C(5,2)*3! = 10*6 = 60
Starts with 1 ends with 6:
C(5,2)*C(3,2) = 10*3 = 30
Starts with 2 ends with 2 or 6:
2*C(5,2)*3! = 2*10*6 = 120
120+60+30 = 210
The notation C(5,2) reads a "5 choose 2" and is the number of combinations when taking two items from a collection of five. It is calculated as
5!/(2!*(5-2)!) =
(5*4*3*2*1)/((2*1)*(3*2*1)) =
(5*4)/(2*1) = 20/2 = 10
In the above calculations, the five things are the internal digit places and the two things are the two fives.
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