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How many even numbers less than 600 can be made from the digits: 3,3,4,8,9 with each only being used once. I can't figure out what to do for the 3rd case where 3 digits are needed

Joe Jewels
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  • I'm assuming this is a homework question? The way to proceed is to begin by fixing your last digit, which must be even. So, there are precisely two ways to fix your last digit. Then, an approach you might consider is to choose some number of the four remaining digits to make up the remaining portion of your number. Be sure when you do this to keep track of the fact that you sometimes will use two 3s. (There are a few ways to approach this.) – Michael Jarret Oct 09 '15 at 18:45
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    I ended up getting 2(for 1 digit) + 6(for 2 digits) + 4 (for 3 digits ending with 8) + 2(for 3 digits ending with 8)= 14 possible ways, can anyone confirm this is the correct answer? – Joe Jewels Oct 09 '15 at 19:42
  • @JoeJewels You should edit your question to include what you did rather than leaving your work in the comments where your work might be overlooked. – N. F. Taussig Oct 10 '15 at 08:57

2 Answers2

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We wish to find how many even numbers less than $600$ can be formed from the digits $3, 3, 4, 8, 9$ if each digit is used at most once.

Since the number is even, the units digit of each number must be $4$ or $8$.

One-digit numbers: The only possibilities are $4$ or $8$, giving us two possibilities in this case, as you found.

Two-digit numbers: If the units digit is $4$, then the tens digit can be $3$, $8$, or $9$. If the units digit is $8$, then the tens digit can be $3$, $4$, or $9$. Hence, there are six possibilities, as you found.

Three-digit even numbers: If the units digit of the even number less than $600$ is $4$, the hundreds digit must be $3$. This leaves us with three choices for the tens digit, namely $3$, $8$, or $9$. Hence, we can form three three-digit even numbers less than $600$ with units digit $4$ by using the digits $3, 3, 4, 8, 9$ at most once. They are $334$, $384$, $394$.

If the units digit of the three digit even number less than $600$ is $8$, we have two possibilities for the hundreds digit, namely $3$ or $4$. If the hundreds digit is $3$, we have three possibilities for the tens digit, namely $3$, $4$, or $9$. If the hundreds digit is $4$, we have two possibilities for the tens digit, namely $3$ or $9$. Thus, we can form five three-digit even numbers less than $600$ with units digit $8$. They are $338$, $348$, $398$, $438$, $498$.

Hence, there are a total of eight three-digit even numbers less than $600$ that can be formed with the digits $3, 3, 4, 8, 9$ if each digit is used at most once.

In all, we can form $2 + 6 + 8 = 16$ even numbers less than $600$ using the digits $3, 3, 4, 8, 9$ at most once.

N. F. Taussig
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That looks straight forward. In order to be even, the number must end in 4 or 8. If it ends in 4 then the first 4 digits must be 3, 3, 8, 9. If there were 4 distinct digits there would be 4!= 24 different orders but because two of those digits are "3", each of those orders is exactly the same as another with just the "3"s swapped so there are really 4!/2= 24/2= 12 such numbers. If the number ends in 8 then the first four digits must be 3, 3, 4, 9 and exactly the same argument applies.

user247327
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