permutation?

Question

1) How many different numbers of 4 digits may be formed with the digits 0,1,2,5,6,7,8 if no digit is used more than once in any number?
2) How many of the numbers formed in question #1 are even?
3) Find the sum of all the numbers formed in question #1?

Answer 1

1) The first question is can the first digit be 0? I'll assume yes, then you have 7 choices for the first, 6 choices for the second, 5 choices for the third and 4 for the fourth. 7 x 6 x 5 x 4 = 840
2) The last digit can be any of the seven with equal probability. 4/7 of the time this will be an even number. 840 * 4/7 = 480
3) There are 840 numbers and each digit will be represented evenly. 0 + 1 + 2 + 5 + 6 + 7 + 8 = 29. So the average digit is 29 / 7. The average of each number will be 29/7 * (10^3 + 10^2 + 10^1 + 10^0)
Therefore the sum will be 840 * (29 / 7 * 1111) = 3,866,280

Answer 2

1) the things mentioned above are not fully correct, because remember 0 can't not be the first digit,so 6x6x5x4 is = 720.

2) chances for the odd number, 1,5 and 7 to be the first digit is 1/2 of 720, which is 360. and to fill in the rest of the digits, u have 2 odd numbers, and 4 even numbers, which gave a chance of 2/3 of getting even number, 360 x 2/3 = 240

chance for even number, 2, 6, and 8 is also one half, which is 360. the remaining digits have 3 odds 3 even, which make it 1/2 of the chances of getting even, 360 x1/2 = 180

240+180= 420

420 of those number are even

3) 1, 2, 5, 6, 7, 8 are the only available number for the thousand unit.

if 1 is the first numbers: the sum of the remaining number are:
28, 2+5+6+7+8+0=28
if 2 is the first numbers: 27
if 5 is the first numbers: 24
if 6 is the first numbers: 23
if 7 is the first numbers: 22
if 8 is the first numbers: 21

so the average of the sum of the remaining digits (hundred, ten and one) is
(29+27+24+23+22+21)/(6x6) = 145/36
the average of the thousand digit is
(1+2+5+6+7+8)/6 =29/6
so the sum of all the numbers formed are
720 x (1000 x 29 / 6 + 100 x 145 / 36 + 10 x 145 / 36 + 1 x 145 / 36) = 3801900

Answer 3

The first given answer is wrong. For a few reasons. First off, it's not 7 choose 4. 7 choose 4 is looking for a set of 4 from 7 that's unordered. You are making 4 digit numbers where order matters (i.e. 2158 is different than 5812). You are looking for the number of ordered sets... I give hints below. (He just edited his answer actually so ignore that..)

1) The first digit has 7 choices given, the second has 6, 3rd has 5...
2) The numbers must end in an even number.
3) All numbers will be in all of the positions an equal amount of times. Remember that a 4 digit number in base 10 is equal to:

(4thDigit)*10^3+
(3rdDigit)*10^2+
(2ndDigit)*10^1+
(1stDigit)*10^0

Regard each digit as an equivalent sum...

Answer 4

1.) I know that if you take all the numbers times and count backwards 4 in this instance you will find the answer. (Ex. 7*6*5*4= 840
2.) Unfortunately I'm not going to work this one out for there are 840 different answers.
3.) I'm pretty sure 840 is the anser to this one!
HAPPY TO HELP!!

Answer 5

5!=120