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1. The parabola is y = 1/4*x^2 - 3x + 9, so a + b + c = 1/4 + (-3) + 9 = 25/4.

2. The equation of the parabola is y = 2x^2 - 4x + 7, so a + b + c = 5.

a=2;c=0; b=a * 2^a;c=c+b; if(c==2^(a+10), goto5, goto6);printc, a, b; a++;if(a<2000, goto2, 0)

OUTPUT = n = 513.               2. 2^2 + 3. 2^3 + 4. 2^4 +............+ 513 * 2^513 = 2^(513 +10)

Alright, the answer is 3276. Without any restrictions, there a total of  ways that the three people can be seated. We must then subtract the cases where at least two people are adjacent. There are two possible cases.

Case 1. All three people are sitting next to each other, in one block.

There are 18 ways to choose three consecutive seats. There are then 3! = 6ways to place the three people, which gives us 18 * 6 =108 possible seatings.

Case 2. There are two people sitting next to each other, and the third person is not next to either of them.

There are 18 ways to choose two consecutive seats. There are then 14 ways to choose the third seat. After the three seats have been chosen, there are  ways to place the three people, which gives us 18*14*6=1512 possible seatings.

Therefore, there are 4892-108-1512 = 3276 seatings where no two people are adjacent.

Thank you all for the effort, it's greatly appreciated!

Well, isn't there 18*17*16 ways to pick? 18 choices for the first person, etc... thats 4896. Could you explain why there are 18*14 cases of 2 people adjacent? Thanks again!

There are 18*17*16 ways to choose the three seats, if people can be adjacent.  We must subtract the cases where there are two people adjacent, which is 18*13*6.  So the number of ways is 18*17*16 - 18*13*6 = 3492.

Let me read over your work, and check. Thanks for your effort, I appreciate it! Thanks!

Total number of ways people can be seated:

18 choose 3

\(\frac{18!}{3!15!}=816\)

Since no two people can be adjacent we have two cases

Three people adjacent

18 ways (18 choose 1)

Two people adjacent (18 * 14)

So 816 - 18 - (18*4)

546

Is this right?

One leg of a right triangle measures 3 cm. The hypotenuse of the right triangle measures 3 cm less then twice the length of the unknown leg. Find the measure of all of the sides of the triangle.

sides are   3,  4,  5    

We know that P(a) - P(b) is divisible by a - b.  This means if a - b is not a prime number, then P(a) - P(b) is not a prime number.  So by modular arithmetic, we can find a positive integer n such that P(n) is composite.

If P(a_1) and P(a_2) are composite, then P(a_1) - P(a_2) will be divisible by some prime number.  Similarly, if P(a_1) - P(a_2) is divisible by some prime number, and P(a_1) or P(a_2) is composite, then the other one is also composite.

We can then take a_1 = n.  So there exists a positive integer a_2 such that P(a_2) is composite.  But since P(a_2) is composite, we can apply the result above, to find a positive integer a_3 such that P(a_3) is composite.  In this way, we can generate an infinite number of positive integers n such that P(n) is composite.

We know that P(a) - P(b) is divisible by a - b.  This means if a - b is not a prime number, then P(a) - P(b) is not a prime number.  So by modular arithmetic, we can find a positive integer n such that P(n) is composite.

If P(a_1) and P(a_2) are composite, then P(a_1) - P(a_2) will be divisible by some prime number.  Similarly, if P(a_1) - P(a_2) is divisible by some prime number, and P(a_1) or P(a_2) is composite, then the other one is also composite.

We can then take a_1 = n.  So there exists a positive integer a_2 such that P(a_2) is composite.  But since P(a_2) is composite, we can apply the result above, to find a positive integer a_3 such that P(a_3) is composite.  In this way, we can generate an infinite number of positive integers n such that P(n) is composite.

I hat you

Forget  answer #2 !!.

thank you!

sumfor(n,2, 17,n*2^n =4194304 =2^(12 + 10)

By arithmetico-geometric series, the series on the left is 2^n*(n - 2).  Then n - 2 = 2^10 = 1024, so n = 1026.

bruh moderation

leg 1 = 3

leg 2 = x

hypotenuse = 2x - 3

set up equation using pythagorean theorem

3² + x² = (2x-3)²

now solve it

you should get an integer answer for x

There are actually 16 triangles of the first kind, so the answer is 16 + 4 = 20.

Thanks for the right answer!

Since you have at most 2 pennies, you can have 0 pennies, 1 penny, or 2 pennies.

The number of ways to select 0 pennies and 4 nickels:  ₅C₀ · ₆C₄

The number of ways to select 1 penny and 3 nickels:     ₅C₁ · ₆C₃

The number of ways to select 2 pennies and 2 nickels:  ₅C₂ · ₆C₂

Add those three answers to together and divide by the number of ways to select 4 coins from 11 coins:

The number of ways to select 4 coins from 11 coins:     ₁₁C₄