# How many even numbers of four digits can be formed with the digits 0,1,2,3,4,5 and 6 no digit being used more?

My attempt to solve this problem is:

First digit cannot be zero, so the number of choices only $6 (1,2,3,4,5,6)$

The last digit can be pick from $0,2,4,6$, so the number of choices only 4

Second digit can be only pick from the rest, so the number of choices only 5

Third digit can be only pick from the rest, so the number of choices only 4

The total number of choices is $6\cdot 4\cdot 5\cdot 4= 480$

So, is my solution true? Or I miss something? Thanks

• If 2,4, or 6 is used for the first digit, can it also be used as the last digit? Oct 1, 2013 at 16:55
• Since the requirement for an even number is that the last digit should be an even number, that's what you want to start with. There on calculate available choices for other places. Jan 15, 2015 at 0:37

For an even no. last digit needs to be even = 4 choices

Total 4 digit even numbers = (Total even numbers) - (Total 3 digit even numbers)

Total even numbers: $4\cdot 6\cdot 5\cdot 4 = 480$

Total 3-digit even numbers: $3\cdot 5\cdot 4\cdot 1 = 60$

Total 4-digit even numbers: $480 - 60 = 420$

• Your last line should read $480 - 60 = 420$. Jan 15, 2015 at 0:46
• whoops. Thanks for notifying. Edited! Jan 15, 2015 at 0:47

You are missing that you might have only three choices for the last digit, depending on your choice of the first digit. You should split into two cases: first digit even and first digit odd. Then your approach goes through nicely. By the way, it is number of choices, not probability.

• Thanks for your correction, I've just changed the 'probability' with 'number of choices' Oct 1, 2013 at 16:58
• First digit even: 3 choices, last digit: 4 choices, so the choices are: 3*4*5*4= 240. First digit odd: 3 choices, so the choices are 240 too. Total choices are still 480. Am I still missing something? Oct 1, 2013 at 18:26
• No, if the first digit is even, there are only three choices for the last digit, because you used one up already. Oct 1, 2013 at 18:34

Consider two cases: Case 1: first digit even There are three ways to find the first digit, if the first digit is even. There are 2,4, and 6. There are three ways to find the last digit, because one is taken for the first digit. There are 5 ways to find the second digit and 4 ways to find the third digit. So, the total ways is: 3*5*4*3= 180 ways

Case 2: first digit odd There are three ways to choose the first digit, there are: 1,3,5 There are 4 ways to choose the last digit, there are 0,2,4,6 There are 5 ways to choose the second digit and 4 ways to choose the third digit. So, the total ways is: 3*5*4*4= 240 ways

Total ways: case 1+ case 2= 180+240= 420 ways

• Here's another way to do this. Consider counting the four digit numbers naively as you first did: There are $4$ choices for the last digit being even. Then there are $6$ remaining choices for the second last, $5$ choices for the third and $4$ choices for the fourth. This gives $4\times 6\times 5\times 4 = 480$. But this also counts $4$ digit even numbers beginning with a $0$, i.e. $3$ digit numbers formed from your selection of numbers without using a $0$. By a similar argument, there are a total of $3\times 5 \times 4 = 60$ of these. So you have all in all $480 - 60 = 420$ choices.
– EuYu
Nov 1, 2013 at 16:44

Consider cases:

1. The last digit is $0$. Then you have six choices for the first digit, five choices for the second digit, and four choices for the third digit, giving $6 \cdot 5 \cdot 4 \cdot 1 = 120$ four digit even numbers that end in $0$.

2. The last digit is $2$, $4$, or $6$. Since the first digit cannot be zero, you have five choices for the first digit, five choices for the second digit, and four choices for the third digit, giving $5 \cdot 5 \cdot 4 \cdot 3 = 300$ four digit even numbers that end in $2$, $4$, or $6$.

That exhausts the possibilities, so there are $120 + 300 = 420$ even four digit numbers that can be formed using the digits $0, 1, 2, 3, 4, 5, 6$.

Make two cases for this problem

Case 1: fix 0 in last position, so last position can be filled in 1 way. Then first place can be filled in 6 ways, second place can be filled in 5 ways and son on, so total of this case will be 6x5x4x1 = 120

Case 2: Since all even numbers (2,4,6) will come is last position, it can be filled in 3 ways, then first place can be filled by 3 odd numbers + 2 even numbers (since one even number is fixed in last position), second position can be filled by 4 digits + 0 that is 5 digits, third place can be filled by 4 digits and last place by 3, so total of this case will be 5x5x4x3 = 300

total required numbers = case 1 + case 2 = 120+300 = 420.

Without considering different cases.

Here we start by finding the total amount of the four-digit numbers with distinct digits, then finding the amount of odd digits, filling the same criteria, and last, we subtract the odd numbers from the total, to find the even numbers filling the criteria.

We have totally seven digits. We know that the digit zero cannot be placed at the position representing the thousand position. That leaves us with six digits to choose from for this position and that can be made in

$$P(6,1) = \frac{6!}{(6-1)!} = \frac{6!}{5!}=6$$, different ways.

Now, we have three positions left to fill and six digits to choose from, including the digit zero, which can be placed anywhere in the remaining positions. This choice can be done in

$$P(6,3) = \frac{6!}{(6-3)!} = \frac{6!}{3!}=6 \cdot5 \cdot4$$, different ways.

Finally, with distinct digits, there is

$$6 \cdot6 \cdot5 \cdot4 =720$$

four-digit numbers to be constructed.

We know that the amount of odd numbers plus the amount of even numbers equal the total amount of the 720 four-digit numbers.

The odd numbers are 1, 3 and 5. The question to be asked is how many of the 720 are odd?

The digit to fill the unit position can only be chosen from the digits 1, 3 or 5, and this choice can be made in

$$P(3,1)=\frac{3!}{(3-1)!}=\frac{3!}{2!}=3$$, different ways.

To choose the digit filling the thousand position, we have five valid digits to choose from, since the zero digit is not valid for this position. The choice can be made in

$$P(5,1)=\frac{5!}{(5-1)!}=\frac{5!}{4!}=5$$, different ways.

Now we are left with two positions to fill, the hundred and tenth position, and five digits to choose from, now including the zero digit. The choice for these two positions can be made in

$$P(5,2)=\frac{5!}{(5-2)!}=\frac{5!}{3!}=5 \cdot4=20$$, different ways.

The total number of odd four-digit numbers, with distinct digits are

$$5 \cdot5 \cdot4 \cdot3 =300$$.

Now, we can answer the question how many of these 720, four-digit numbers, are even, by the subtraction

$$720-300=420$$.

The even numbers are 420.