How many ways are there to put 9 differently colored beads on a $3\times3$ grid if the purple bead and the green bead cannot be adjacent (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same

20,160

Step-by-step explanation:

The arrangement of the 9 differently colored bead can be presented as follows;

Where 1 is the purple bead and 2 is the green bead, the number of ways of arrangement where the green bead cannot be adjacent to the green either horizontally, vertically, or diagonally

Placing the purple bead at 1, the location of the green bead = 3, 6, 7, 8, or 9

The number of ways = 5 ways × 7! ways of arranging the other beads

With the purple bead at 2, the location of the green bead = 7, 8, or 9

The number of ways = 3 × 7!

With the purple at 3, we also have 5 × 7! ways

At 4, similar to 2, we have, 3 × 7! ways

At 5, we have, 0 × 7!

At 6, we have 3 × 7!

For 7, 8, and 9, we have, (5 + 3 + 5) × 7!

The total number of ways = (5 + 3 + 5 + 3 + 0 + 3 + 5 + 3 + 5) × 7! ways

However, placing the purple bead at 1, 2, 3, 4, 6, 7, 8, and 9, (8 positions) can be taken as reflection and rotation of each other and can be considered the same

Therefore, the total number of acceptable ways = (5 + 3 + 5 + 3 + 0 + 3 + 5 + 3 + 5) × 7!/8 = 20,160 ways

i think it would be 43 movies that would have to be watched.

step-by-step explanation:

think about it. for the second plan, every month is 8 dollars. 8 x 12= 96. so thats the number for the second plan. now for the first plan, its annual meaning you have to pay 10 dollars every year. so now you have to get an equivalent amount of 86 since 96-10=86. 86 divided by 2 since every movie costs 2$ then you would have to watch 43 movies a year to get the same as the second plan. i really hoped this . have a wonderful day as well!

step-by-step explanation:

always plug in x. in this case, x equals 4.

i'm going to first solve 4(2+3x)

1) 4(2+3(4)) use pemdas to solve.

i'm going to multiply whats in the parenthesis first, so 3 times 4=12.

2)4 (2+12) i'm then going to multiply 4 by both numbers in the parenthesis (because multiplying comes before adding)

4 times 2= 8.   4 times 12= 48

4) (8+48)= 56 (answer)

i'm then going to solve 8+12x and plug in 4 for x.

1) 8 +12(4)

multiplying comes before adding, so i'm gonna multiply 12 and 4.

12 times 4= 48.

2) 8+48=56

both equal the same!