How many numbers of four different digits each can be
formed from the digits 0,2,3,5,6,9?
There are 5 ways to choose the 1st digit. (can't choose 0)
There are 5 remaining ways to choose the 2nd digit. (can choose 0)
There are 4 remaining ways to choose the 3rd digit.
There are 3 remaining ways to choose the 4th digit.
answer: 5∙5∙4∙3 = 300 ways.
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Of these numbers, how many are even?
Case 1. 0 is chosen for the 4th digit.
There is 1 way to choose the 4th digit. (as 0)
There are 5 remaining ways to choose the 1st digit.
There are 4 remaining ways to choose the 2nd digit.
There are 3 remaining ways to choose the 3rd digit.
answer for case 1: 1∙5∙4∙3 = 60 ways.
Case 2. 0 is not chosen for the 4th digit.
There are 2 ways to choose the 4th digit. (as 2 or 6)
There are 4 remaining ways to choose the 1st digit. (can't choose 0)
There are 4 remaining ways to choose the 2nd digit.
There are 3 remaining ways to choose the 3rd digit.
answer for case 2: 2∙4∙4∙3 = 96 ways.
Total for the two cases: 60+96 = 156
how many are divisible by 5?
Case 1. 0 is chosen for the 4th digit.
Same as case 2 in the preceding problem:
answer for case 1: 60 ways.
Case 2. 5 is chosen for the 4th digit.
There is 1 way to choose the 4th digit. (as 5)
There are 4 remaining ways to choose the 1st digit. (can't choose 0)
There are 4 remaining ways to choose the 2nd digit.
There are 3 remaining ways to choose the 3rd digit.
answer: 1∙4∙4∙3 = 48 ways.
Total for the two cases: 60+48 = 108 ways.
Edwin
