Answer:
case 1 = 2592
case 2 = 729
case 1 + case 2 = 2916
(this is not a direct adition, because case 1 and case 2 have some shared elements)
Step-by-step explanation:
Case 1)
6 digits numbers that can be divided by 25.
For the first four positions, we can use any of the 6 given numbers.
For the last two positions, we have that the only numbers that can be divided by 25 are numbers that end in 25, 50, 75 or 100.
The only two that we can create with the numbers given are 25 and 75.
So for the fifth position we have 2 options, 2 or 7,
and for the last position we have only one option, 5.
Then the total number of combinations is:
C = 6*6*6*6*2*1 = 2592
case 2)
The even numbers are 2,4 and 6
the odd numbers are 3, 5 and 7.
For the even positions we can only use odd numbers, we have 3 even positions and 3 odd numbers, so the combinations are:
3*3*3
For the odd positions we can only use even numbers, we have 3 even numbers, so the number of combinations is:
3*3*3
we can put those two togheter and get that the total number of combinations is:
C = 3*3*3*3*3*3 = 3^6 = 729
If we want to calculate the combinations togheter, we need to discard the cases where we use 2 in the fourth position and 5 in the sixt position (because those numbers are already counted in case 1) so we have 2 numbers for the fifth position and 2 numbers for the sixt position
Then the number of combinations is
C = 3*3*3*3*2*2 = 324
Case 1 + case 2 = 324 + 2592 = 2916
![z= \rho [\cos \theta + i \sin \theta]](https://tex.z-dn.net/?f=z%3D%20%5Crho%20%5B%5Ccos%20%5Ctheta%20%2B%20i%20%5Csin%20%5Ctheta%5D)
, can be calculated as follows:![z^n = \rho^n [\cos n\theta + i \sin n\theta ]](https://tex.z-dn.net/?f=z%5En%20%3D%20%5Crho%5En%20%5B%5Ccos%20n%5Ctheta%20%2B%20i%20%5Csin%20n%5Ctheta%20%5D)
is equal to![z^3 = \rho^3 [\cos 3 \theta + i \sin 3 \theta ] = (5^3) [\cos (3 \cdot 330^{\circ}) + i \sin (3 \cdot 330^{\circ}) ]](https://tex.z-dn.net/?f=z%5E3%20%3D%20%5Crho%5E3%20%5B%5Ccos%203%20%5Ctheta%20%2B%20i%20%5Csin%203%20%5Ctheta%20%5D%20%3D%20%285%5E3%29%20%5B%5Ccos%20%283%20%5Ccdot%20330%5E%7B%5Ccirc%7D%29%20%2B%20i%20%5Csin%20%283%20%5Ccdot%20330%5E%7B%5Ccirc%7D%29%20%5D)
(1)
, (1) becomes![z^3 = 125 [\cos 270^{\circ} + i \sin 270^{\circ} ]](https://tex.z-dn.net/?f=z%5E3%20%3D%20125%20%5B%5Ccos%20270%5E%7B%5Ccirc%7D%20%2B%20i%20%5Csin%20270%5E%7B%5Ccirc%7D%20%5D)