Consider the lists of length six made with the symbols P, R, O, F, S, where repetition is allowed. (For example, the following is such a list: (P, R,O, O,F, S).) How many such lists can be made if the list must end in an S and the symbol O is used more than once?

96

Step-by-step explanation:

COMBINATIONS & PERMUTATIONS can be confusing but they always have a solution.

Key 1:  Understand and/or rewrite the question

Consider the lists of 6 items (out of the 5 letters P,R,O,F, and S; where O is used twice).

NOTE: When they say O is used more than once, don't forget that each list must not exceed 6 items in total. So, O is used twice. Simple!

How many such lists can be made if each list must end in an S?

NOTE: This instruction requires that S doesn't move. It won't change position, and it won't be involved in the rearrangement!

Key 2:  Solve the mysteries

Now, S won't move and O is doubled. These rules put a form of restriction on the "n" which is the total number of items involved in the operation. We are supposed to create lists of 6 items but the reality is - only 5 are moving. Recall also that the original number of items involved in the operation is 5.

In this case, n = 5

Then we have 4 items rotating - P, R, O and F - with 1 repeated.

What this means is that for every list starting with the first O, the different arrangements apply to the second O as well.

There is no distinction such as O₁ and O₂. O is O!

Let k = the extra O

(n - k)  =  5 - 1  = 4

So each of these 4 letters has 4! (4 factorial) arrangements, that is (4 x 3 x 2 x 1) = 24 arrangements

Multiply 24 by 4 to get 96

4038

step-by-step explanation:

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the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is