Imagine you have N chess pieces and M squares to put them, and M > N. Precisely how many different arrangements are there, if each square can only be occupied by one piece and all the pieces are identical and cannot be distinguished?

Let's consider a numerical example.

Let's say we had N = 4 chess pieces that are all pawns and there are M = 9 squares to place them. Label the squares A,B,C,D,E,F,G,H,I

Now because all the pieces are identical, this means that what we can effectively do is select four letters at random from the set {A,B,C,D,E,F,G,H,I} and this basically will instruct where to place the pawns. Whatever letter is selected will have its space filled with a pawn, otherwise, the location remains empty.

For instance, if we select the subset {D,G,A,C} then this means to place pawns at squares D, G, A and C. The order of placement does not matter because again, we can't tell the pawns apart. The letters that are not selected are simply left empty (with no chess piece to occupy the location).

As you can probably guess or see at this point, we will use the nCr combination formula to count how many ways to fill the 9 squares using only 4 pieces.

In this particular example, n = 9 and r = 4. There are...

C(n,r) = C(9,4) = (9!)/(4!*(9-4)!) = (9!)/(4!*5!) = (9*8*7*6*5!)/(4!*5!) = (9*8*7*6)/(4*3*2*1) = 3024/24 = 126 ways to pick four locations for the pawns to be situated. This only applies to a 9 square chessboard (perhaps a 3 by 3).

Now onto a more general expression

we have N pieces and M locations, with M > N

Again we turn to a combination formula

Instead of C(n,r), we compute C(M, N), so

C(M, N) = (M!)/(N!*(M-N)!)

We cannot simplify further because we don't know the values of M or N. Back in the example in the previous section, we had M = 9 and N = 4 which computed to 126.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

answer: no

step-by-step explanation:

you are only given the cost not the amount of pounds.