Respuesta :
The total number of different colorings possible are c)3120.So, correct option is c.
We know very well that Probability is the defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.
We need to take care that each vertex of convex pentagon has been assigned a color.
We are given that we can choose 6 colors , and we need to take care that the ends of each diagonal must have different colors.
So, we need to try all different combinations,
Let PQRST are the vertex of the pentagon.
Let vertex P be any vertex, then vertex Q be one of the diagonal vertices to P, R be one of the diagonal vertices to Q, and so on. We consider cases for this problem.
In the case that,
R has the same color as P,
S has a different color from P
and so,
T has a different color from P and S.
In this case, the choices are
P = 6
Q = 5
R = 1
S = 5
T = 4
So, total number of combinations are =6 x 5 x 1 x 5 x 4 = 600 combinations.
Similarly, R has a different color from P and
S has a different color from P,
In this kind of situations,
P has 6 choices,
Q has 5 choices,
R has 4 choices
S has 4 choices,
and T has 4 choices,
So, total number of combinations is
=> 6 x 5 x 4 x 4 x 4 = 1920
Next case, R has a different color from P and S has the same color as P, So,
P has 6 choices,
Q has 5 choices,
R has 4 choices,
S has 1 choice, and
T has 5 choices,
=> total number of combinations=6 x 5 x 4 x 1 x 5 = 600
On Adding all those combinations up, we get
=> 600 + 1920 + 600 = 3120
Hence, total colorings are c)3120.
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(Complete question) is:
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
A.2520 B.2880 C.3120 D.3250 E.3750
