Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)

The probability that there is at least one pair but not a three of a kind is 0.6944.

Step-by-step explanation:

We have the case when we have five dices and we roll them and we need  that there is at least one pair but not a three of a king which is Two dice showing the same value, but no three dice showing the value.

Total no of outcomes =

We will consider the other cases and then subtract them from the total outcomes.

No of outcomes of 5 of a kind = 6

No of outcomes 4 of a kind and 1 different = (6*5)*5 = 150

No of outcomes with 3 of a kind and 1 different pairs = (6*1*1*5*1)*5C2= 300

No of outcomes with 3 of a kind and two different = 6*5*4*10= 120*10=1200

No of outcomes when all are different = 6! = 720

These all the cases when subtracted from the total will represent the total cases in which there are two dice showing the same value.

Total cases = 6+150+300+1200+720

                   = 2376

Required no of outcomes = 7776-2376 = 5400

Probability = = 0.6944

Therefore the probability for the case of at least one pair but not a three of a kind is 0.6944.

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

i think its a and d

step-by-step explanation: