SOLUTION: How many three digit even numbers less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6, and 7 if the integers cannot be used more than once?

Algebra ->  Permutations -> SOLUTION: How many three digit even numbers less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6, and 7 if the integers cannot be used more than once?      Log On


   

Question 634813: How many three digit even numbers less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6, and 7 if the integers cannot be used more than once?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
There are four scenarios:

Scenario A)

You choose 2 even numbers for the hundreds and tens digits. So you'll have one even number left over (since there are only 3 even numbers in this list). So there are 3 ways to play out this scenario

Since something like 246 is different from 426, this means that we double the count to 2*3 = 6

BUT

we don't count something like 642 or 624 since we can't go over 500.

So the count so far is 6 - 2 = 4
--------------------------------

Scenario B)

You chose one even number for the hundreds digit. Then one odd number for the tens digit. Finally you choose one even number for the units digit.

So you have 2 choices (2 or 4) for the hundreds digit.

Then you now have 4 choices (1,3,5,7) for the tens digit

Finally, you have 2 choices for the units digit (2, 4, 6, but kick out either 2 or 4 -- the number you chose for the hundreds digit)

So add on 2*4*2 = 16

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Scenario C)

You choose an odd number for the first digit and an even number

So you have 2 choices for the hundreds digit (1, 3)

Then you have 3 choices for the tens digit (2, 4, 6)

Finally, you have 2 choices for the units digit (2, 4, 6, but eliminate what you chose for the tens digit)

So add on 2*3*2 = 12

-------------------------------
Scenario D)

The hundreds and tens digit are both odd.

So you have 2 choices for the hundreds digit (1 or 3), then you have 3 choices for the tens digit (1, 3, 5, 7...but take off one number), finally you have 3 even digits to choose from

So you add on 2*3*3 = 18 numbers

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So the total number of ways to do this is

4+16+12+18 = 50

Therefore, there are 50 ways to make an even 3 digit number that is less than 500 using numbers from this list {1, 2, 3, 4, 5, 6, 7} where each number is only used once.


Here is the list of the 50 numbers:

124
126
132
134
136
142
146
152
154
156
162
164
172
174
176
214
216
234
236
246
254
256
264
274
276
312
314
316
324
326
342
346
352
354
356
362
364
372
374
376
412
416
426
432
436
452
456
462
472
476

Again, there are 50 numbers in that list above