急! F5排列與組合 4q1

Case 1: the units digit is 0.
There are 9 x 8 x 7 ways, ie 504.

Case 2: the units digit is 5.
As the thousands digit cannot be 0. So there only 8 x 8 x 7 ways, ie. 448.

So, totally there are (504 + 448) ways, ie. 952 ways. (ans.)


First of all, there are 9 x 9 x 8 x 7 x 6 ways to form a 5-digit number, ie 27216.
Now, try to calc. How many 5-digit number that are divisible by 10.
Same as the previous question, as only 0 can be the units digit,
there are only (9 x 8 x 7 x 6) ways, ie 3024 ways.
So there are only (27216 - 3024) numbers that are not divisible by 10 can be formed, ie. 24192. (ans.)
(as number is formed by the cards, so all digits are different.)

1. By choosing 4 different numerals from the ten numerals 0 to 9, how many 4-digit numbers that are divisible by 5 can by formed ?
First, the thousand place cannot be 5 or 0.
∴Total number of numbers suitable is 10-2=8.
Second, the hundred place cannot be 5, 0 or the first number.
∴Total number of numbers suitable is 8-1=7.
Third, the ten place cannot be 5, 0 the first number or the second number.
∴Total number of numbers suitable is 7-1=6.
Finally, the one place can be only 5 or 0 since it must be divisible by 5.
∴Total number of numbers suitable is 2.

∴Total number of 4-digit numbers that are divisible by 5
=8x7x6x2
=672

2. By using 10 cards marked 0 to 9 respectively, how many 5-digit numbers that are not divisible by 10 can be formed ?
Using the method in q.1, you can easily find out the answer. However, there are some special cases that I need to tell you.
First, the ten-thousand place cannot be 0.
∴Total number of numbers suitable is 10-1=9.
Second, the thousand place can be 0.
∴Total number of numbers suitable is 9-1+1=9.
Third, the hundred place.
Total number of numbers suitable is 9-1=8.
Fourth, the ten place.
Total number of numbers suitable is 8-1=7.
Finally, the one place cannot be 0 since it is not divisible by 10.
∴Total number of numbers suitable is 7-1-1=5.

∴Total number of 5-digit numbers that are not divisible by 10
=9x9x8x7x5
=22680
NOTE: Although the question doesn't say that you can use the cards once only, you must notice that the cards can be used once only since there are only 10 cards.