AvenJohn

x^2-6x+k=0

Factor the equation into (x+a)(x+b)=0

We know that a+b=-6 and the maximum value of ab is 9

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There are b-1 choices for the first digit, b-1 choices for the second, and b-2 choices for the third. That means (b-1)(b-1)(b-2)=180 so b=7.

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Let's use complementary counting. The only way for no girls to be next to each other is if they sit b-g-b-g-b-g-b. There are 4!=24 ways to arrange the boys and 3!=6 ways to arrange the girls in this case. That makes 144 cases. There are 7!=5040 ways to organize the people with no restrictions so the answer is 5040-144=4896.

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Because the circumcenter is the intersection of perpendicular bisectors, we can graph two perpendicular bisectors from two lines. Then we know the circle's center, (-4,2). Once we get the radius, we can use the formula (x-a)^2+(y-b)^2=r^2 where a and b are the coordinates of the circles center.

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So we know (n-2)(n+18)=n(2n-4). Simplifying gets us n^2+16n-36=2n^2-4n so n^2-20n+36=0. We can factor this equation into (n-2)(n-18) so n=18.

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I don't think you quite got it. There are 9*10^4 5 digit numbers and 6*7^4 that have no 2's, 3's, or 4's. Using complementary counting, the answer is 75594.

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There are 6choose2 or 15 ways to paint two faces. There are two ways to get a product of 6. (6,1) and (3,2). 15-2=13

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10*9*8*7*6*5*4*3*2*1 turns into 10/9/8*7*6/5*4*3/2*1 so 4 *'s were replaced.

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You have to go right 8 times and up 2 times. We can write this as RRRRRRRRUU. There are 10!/(8!*2!) ways to arrange it so the answer is 45

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yea... i shouldnt have made the assumption of order. thanks