1. ## counting and permutation

Given the digits 1,2,3,4 and5 find how many 4-digit numbers can be formed from them:

(a)if the number must be even,without any repeated digit
(b)if the number must be even
(c)if repetitions of a digit are allowed

thank you.

3. Originally Posted by calfever
Given the digits 1,2,3,4 and5 find how many 4-digit numbers can be formed from them:

(a)if the number must be even,without any repeated digit
(b)if the number must be even
(c)if repetitions of a digit are allowed

thank you.
I'll start you off, for a) if the number is even it must end in 2 or 4.

Then you would have two cases of a permutation of 3 digits taken from 4 numbers. Like Plato said, attempt the rest, show your work, and you will get excellent help.

4. (a)4*3*2*2
(b)5*5*5*2
(c)5*5*4*3

correct?

5. 1.how many natural numbers greater than or equal to 1000 and less than 5400 have the properties:
(a)no digit is repeated
[my ans.4(9*8*7)+(4*5*4)]
(b)the digits 2 and 7 do not occur
[my ans.3(7*7*7)+1+(3*4*4)

2.how many 6-digit numbers can be formed using {1,2,...,9}with no repetitions such that 1 and 2 do not occor in consecutive positions?
[my ans.(9*8*7*6*5*4)-(7*6*5*4*5)

3.how many positive integers less than 1,000,000 can be written using only the digits 7,8 and9? how many using only the digits 0,8 and9?
[my ans.3^1+3^2+...+3^6] and [my ans.3+2(3+3^2+...+3^5)]

6. Originally Posted by calfever
(a)4*3*2*2
(b)5*5*5*2
(c)5*5*4*3

correct?
Not even close.

Remember, I told you that for a) you had two cases:

First case the number ends in 2 so you have $\displaystyle \frac{4!}{(4-3)!}$
Second case the number ends in 4 so you have $\displaystyle \frac{4!}{(4-3)!}$

When you have two mutually exclusive cases for one event you add the two together giving 4! + 4!

Part b) is totally different. The numbers can be repeated. So for the first digit you have 5 choices, the second digit you have 5 choices, the third digit you have five choices, but for the fourth digit you only have 2 choices since the number must be even. How would you calculate that?