问题1：假设数字1，数字2，数字3，数字4和数字5可以形成多少个3位数字，
(i)允许重复数字吗?
解决方案：
Answer: 125.
Method:
Here, Total number of digits = 5
Let 3-digit number be XYZ.
Now the number of digits available for X = 5,
As repetition is allowed,
So the number of digits available for Y and Z will also be 5 (each).
Thus, The total number of 3-digit numbers that can be formed = 5×5×5 = 125.
(ii)不允许重复数字吗?
解决方案：
Answer: 60.
Method:
Here, Total number of digits = 5
Let 3-digit number be XYZ.
Now the number of digits available for X = 5,
As repetition is not allowed,
So the number of digits available for Y = 4 (As one digit has already been chosen at X),
Similarly, the number of digits available for Z = 3.
Thus, The total number of 3-digit numbers that can be formed = 5×4×3 = 60.
问题2：如果数字可以重复，那么从数字1、2、3、4、5、6可以形成多少个3位数的偶数?
解决方案：
Answer: 108.
Method:
Here, Total number of digits = 6
Let 3-digit number be XYZ.
Now, as the number should be even so the digits at unit place must be even, so number of digits available for Z = 3 (As 2,4,6 are even digits here),
As the repetition is allowed,
So the number of digits available for X = 6,
Similarly, the number of digits available for Y = 6.
Thus, The total number of 3-digit even numbers that can be formed = 6×6×3 = 108.
问题3：如果没有字母可以重复使用英文字母的前10个字母可以形成多少个4字母的代码?
解决方案：
Answer: 5040
Method:
Here, Total number of letters = 10
Let the 4-letter code be 1234.
Now, the number of letters available for 1st place = 10,
As repetition is not allowed,
So the number of letters possible at 2nd place = 9 (As one letter has already been chosen at 1st place),
Similarly, the number of letters available for 3rd place = 8,
and the number of letters available for 4th place = 7.
Thus, The total number of 4-letter code that can be formed = 10×9×8×7 = 5040.
问题4：如果每个数字都以67开头并且没有一个数字出现多次，那么可以使用数字0到9构造多少个5位数电话号码?
解决方案：
Answer: 336
Method:
Here, Total number of digits = 10 (from 0 to 9)
Let 5-digit number be ABCDE.
Now, As the number should start from 67 so the number of possible digits at A and B = 1 (each),
As repetition is not allowed,
So the number of digits available for C = 8 ( As 2 digits have already been chosen at A and B),
Similarly, the number of digits available for D = 7,
and the number of digits available for E = 6.
Thus, The total number of 5-digit telephone numbers that can be formed = 1×1×8×7×6 = 336.
问题5：抛硬币3次，并记录结果。有多少可能的结果?
解决方案：
Answer: 8
Method:
We know that, the possible outcome after tossing a coin is either head or tail (2 outcomes),
Here, a coin is tossed 3 times and outcomes are recorded after each toss,
Thus, the total number of outcomes = 2×2×2 = 8.
问题6：给定5个不同颜色的标志，如果每个信号需要使用2个标志，一个在另一个标志之下，则可以生成多少个不同的信号?
解决方案：
Answer: 20.
Method:
Here, Total number of flags = 5
As each signal requires 2 flag and signals should be different so repetition will not be allowed,
So, the number of flags possible for the upper place = 5,
and the number of flags possible for the lower place = 4.
Thus, the total number of different signals that can be generated = 5×4 = 20.