Explain and prove your answer for each question. (a) How many lists of length 3 can be formed whose components are drawn from {1, 2, 3, . . . , n}? (b) How many lists of length 3 can be formed whose components are drawn from {1, 2, 3, . . . , n} and if we require that the first and second components must be the same? (c) How many lists of length 3 can be formed whose components are drawn from {1, 2, 3, . . . , n} and the components are all distinct? (d) How many lists of length 3 can be formed whose components are drawn from {1, 2, 3, . . . , n} and if we only require that the first and last components must be different?

a) We have n³ possible lists

b) We have n² possible lists

c) We have n³-3n²+2n possible lists

Step-by-step explanation:

a) We have n options for each element of the list. Since we have no restrictions between the elements, then we will have to power n by 3, gibing us a total of n³ possible lists that we can form.

b) Since the fisrt two components must be the same, then we have n possibilities for the first element and after choosing it, we will have only 1 possibility for the second element and then n possibilities for the third one. This gives us n*1*n = n² possible lists.

c) We have n possibilities for the first element, but we have to discard it when we pick the second one, thus we have n-1 possibilities for the second element and with a similar argument we will have n-2 possibilities for the third element. Giving us a total of n*(n-1)*(n-2) = n³-3n²+2n possibilities.

step-by-step explanation:

we are given

since, we have to isolate x on anyone side

so, we can move all x terms on one side

subtract both sides by 2x

add both sides by 9

divide both sides by 4

we get

37/3

step-by-step explanation:

step 1: simplify both sides of the equation.

7x+8(x+

1

4

)=3(6x−9)−8

7x+(8)(x)+(8)(

1

4

)=(3)(6x)+(3)(−9)+−8(distribute)

7x+8x+2=18x+−27+−8

(7x+8x)+(2)=(18x)+(−27+−8)(combine like terms)

15x+2=18x+−35

15x+2=18x−35

tep 2: subtract 18x from both sides.

15x+2−18x=18x−35−18x

−3x+2=−35

step 3: subtract 2 from both sides.

−3x+2−2=−35−2

−3x=−37

step 4: divide both sides by -3.

−3x

−3

=

−37

−3

x=

37

3