C-12.1 Binomial coefficients are a family of positive integers that have a number of useful properties and they can be defined in several ways. One way to define them is as an indexed recursive function, C(n, k), where the "C" stands for "choice" or "combinations." In this case, the definition is as follows: C(n, 0) = 1, C(n, n)=1, and, for 0

Step-by-step explanation:

a)

If we implement the combinations recursively without memoization, then the running time would be exponential in n.

Time Complexity:

If we implement this equation literally, as a recursive program, then the running

time of our algorithm without using memoization, T(n), as a function of n, has the following behavior:

T(0) = 1

T(1) = 1

T(n) = T(n − 1) + T(n − 2).

But this implies that

T(n) ≥ 2T(n − 2) = 2^(n/2).

b)

But if we store the combinations in an array, C[][], then we can instead calculate the combinations, C(n,k), iteratively, as follows:

C[0,0] = 0

C[1,1] = 1

for i = 2 to n do

C[n,k] = C[n-1, i−1] + C[n-1,k]

This algorithm clearly runs in O(n) time, and it illustrates the way memoization

can lead to improved performance when subproblems overlap and we use table

lookups to avoid repeating recursive calls.

yes

step-by-step explanation:

give me the question

\

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is