Four cards labeled A, B, C, and D are randomly placed in four boxes labeled A, B, C, and D. Count the number of elements in the event that no box contains a card with the same letter as the box.

There are total of 23 events in which the box DO NOT get the SAME label card as the box.

Step-by-step explanation:

Here, the total number of boxes = 4

A , B , C and D

There are a total of 4 cards with same label A, B , C and D.

Now, the total number of ways, by which the 4 cards can be placed in 4 boxes RANDOMLY as

= 4 x 3 x 2 x 1  = 24 ways

Now, the number of ways, the each box with same number has the EXACT SAME card  

= 1 way  which is A in A , B in B , C in C and D in D

So, the number of ways where  no box contains a card with the same letter as the box

= Total ways - The ways with same number has the EXACT SAME card  

=  24 ways - 1 way  = 23 ways

Hence, there are total of 23 events in which the box DO NOT get the SAME label card as the box.

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: