How many even numbers of at least four digits can be formed using the digits 0,1,2,3 and 5 without repetitions? (Answer is 84)
After you've chosen the last digit and the first 3
digits, you then choose whether to stop there and
have a 4-digit number or whether to put the remaining
1 digit between the 3rd and last digits and have a 5-
digit number.
Since 0 can't come first we have two cases, last digit 0
and last digit 2.
Case 1: Last digit = 0
Choose the last digit 1 way (as 0).
Choose the 1st digit 4 ways.
Choose the 2nd digit 3 ways.
Choose the 3rd digit 2 ways.
Choose whether to stop here and have a 4-digit number,
or whether to put the remaining digit between the
3rd and last digits and have a 5-digit number.
That's 2 choices.
Total for case 1: 1*4*3*2*2 = 48
Case 2: Last digit = 2
Choose the last digit 1 way (as 2).
Choose the 1st digit 3 ways. (Can't choose 0)
Choose the 2nd digit 3 ways. (Can choose 0)
Choose the 3rd digit 2 ways.
Choose whether to stop here and have a 4-digit number,
or whether to put the remaining digit between the
3rd and last digits and have a 5-digit number.
That's 2 choices.
Total for case 2: 1*3*3*2*2 = 36
Grand total: 48+36 = 84
Edwin