急! F5排列與組合 4q1

2012-08-24 10:28 am
請詳細步驟教我計以下二條 :

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 ?

2. By using 10 cards marked 0 to 9 respectively, how many 5-digit numbers that are not divisible by 10 can be formed ?

回答 (2)

2012-08-24 6:56 pm
✔ 最佳答案
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.)
2012-08-24 3:31 pm
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.
參考: me


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