Let the number of Red,Blue and Yellow cards be R, B, Y respectively. Points, $P = R+ 2*R* B+ 3*B*Y$ Lets write this as an function of 2 variables Y,R for various values of B which will determine the coefficients, $P= (1+2*B)R +(3*B) Y.$ It can be seen that for all values of B above 1, The coefficient of Y is greater than the coefficient of R. Hence logically we can conclude that R has to be 0 since we cannot have a maximum value for B=0 or B=1. The task reduces to maximizing the product of B*Y which occurs at (R,B,Y)=(0,8,7) or (0,7,8). On substituting the values we get $P=168.$

The key is to find the biggest point. And the biggest point is when the point of yellow card is maximum and how to maximum the point of yellow card? is to maximum red card.

we got if

$a=red card$

$b=blue card$

$c=yellow card$

$a+b+c=15$

point $a= a$

point $b= b \times a \times 2$

point $c= c \times a \times 3$

so like what i said how to maximum $point a+point b+ point c$ is when card $a$ and card $c$ is $7$ so the number for $b$ must be $1$ (Why 7 is maximum for card a and card c? remember how to find point c! is $a \times c \times 3$ and $7 \times 7 \times 3$ is the biggest for c! why not $7 \times 8 \times 3$ ? because the yellow card must be exist!

$point a+point b+ point c$ $=$ $7+14+147=168$

Red cards give $1 + b$ points. Blue cards can give at most $2r$ points. Yellow cards give $3b$ points. Clearly, the points are maximized when there are a certain number of blue and yellow cards such that there are enough blue cards to make yellow cards worth many points, but at the same time enough yellow cards to multiply that gain. Thus, there must also be no red cards.

If $x$ is the number of yellow cards, this can be modeled as $x(3(15 - x))$ . In an expanded form, this is $-3x^2 + 45x$ . This is simply a parabola whose maximum is at its vertex. The x-coordinate of the vertex can be found by $\frac{-b}{2a}$ , which in this case is $7.5$ .

Since a parabola decreases at an equal rate on both sides, the solution is maximized at either $x = 7$ or $x = 8$ . Substituting this in as an answer, we get a maximum point value of $168$ .

include <iostream>

using namespace std;

int main() { int a,b,c,d,e; int max=0; for(a=0;a<=15;a++) { d=15-a; for(b=0;b<=d;b++) { c=15-a-b; e=a+b 2 a+c 3 b; if(max<e) max=e; } } std::cout << max << std::endl;

return 0;

}

R is number of red cards. B is number of blue cards. Y is number of yellow cards. P is point total of cards.

$R + B+ Y = 15 = \phi \; \; \; \; \; P = R + 2RB + 3BY$

Because we have the boundary condition R+B+Y=15 its a good idea to try using a Lagrange multiplier. However, the straight forward method does not give a maximum or minimum. Apparently, without imposing the three remaining boundary conditions, the function P does not have a maximum or minimum. The three remaining boundary conditions are $R \geq 0, \; B \geq 0, \; Y \geq 0$ . I worked the Lagrange method out separately with each of these conditions.

For R = 0

$F = P + \lambda \phi \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\; \; \; \; F = 3BY + \lambda (B+Y)$

$\frac{\partial F}{\partial B} = 3Y + \lambda = 0, \; \; \; \lambda = -3Y$

$\frac{\partial F}{\partial Y} = 3B + \lambda = 0 = 3B - 3Y, \; \; \; 0 = B - Y, \; \; \; B = Y$

$B +Y = 15 = 2B, \; \; \; B = 7.5 = Y$

Since the calculated maximum is between integers the actual maximum will be at one of the two closest integers. B is 7 and Y is 8, or vice versa.

$P = 3BY = 168$

Using the same method with B=0 does not yield a maximum. Using the method with Y=0 gives a maximum of 120. Therefore the maximum score possible with fifteen cards is 168.

Let the number of red cards be x, the number of blue cards be y, the number of yellow cards be (15-x-y).

So the number of points received can be expressed as x+2xy+3(15-x-y)y =-3(y-7.5)^2+675/2+x(1-y) For this value to be maximum, it is obvious that x=0 and y=7 or 8. So the maximum number of points is 7 8 3=168