An ant sits on an infinite grid at (0, 0) as shown below. There is also foodindicated by a dot at (5, 3). The ant is not aware yet of the oblique line(that’s Broadway of the ant city), but she will be in part (e).(a) Assume that the ant senses where the food is and will start moving to-wards it by only making UP and RIGHT moves. How many possible pathsare there that will bring the ant to the food? Hint: think of a bijection fromthe set of paths to some special set of binary words (what is that set?).2

In this version of the story, the ant has no idea where the food is.So she started to move from (0, 0) by making a sequence of random UP,DOWN, LEFT, and RIGHT moves, only to find herself back to her originalposition after 8 moves. Although she was disappointed, she thought: “Howmany possible paths of length 8 will bring me back to my original position?”Well, she did not really think that, but I did. Please count those using thefollowing strategy:• First, divide the paths into disjoint categories, depending on the num-ber of UP moves in the path.• Then observe that the number of UP moves completely determines thenumber of all other types of moves!• Then think of the path as an anagram of length 8 made of U, D, L,and R.• Finally, use the addition rule

An ant named 𝐵B, wants to climb an uneven cliff and reach its anthill (i.e., home of ant). On its way home, 𝐵B makes sure that it collects some food. A group of ants have reached the food locations which are at 𝑥−x−intercepts of the function 𝑓(𝑥)=(𝑥2−21)((𝑥−4)3−1)f(x)=(x 2 −21)((x−4) 3 −1). As ants secrete pheromones (a form of signals which other ants can detect and reach the food location), 𝐵B gets to know the food location. Then the sum of the 𝑥x-coordinates of all the food locations is

find the number of paths that you could take to travel from A and B if you are only allowed to move to the right or down through the grid. You cannot travel a path if the path is not drawn in .

Select the correct answerEach sentence is labelled with a letter. Choose the most logical order of sentences from among the given choices to construct a coherent paragraphA. But when it feels other ants walking on its back, the ant simply stays put.B. When an ant on the march comes to a gap in its path, it slows down.C. In this way, the ants build a bridge long enough to span whatever gap is in front of them.D. The process repeats as the next ant in line slows, gets trampled and freezes in place.E. The rest of the colony, still barreling along at 12 centimeters per second, comes trampling over its back.radio_button_uncheckedBDEACradio_button_uncheckedBEADCradio_button_uncheckedBEDCAradio_button_uncheckedBADEC

The ant crawled ___________ the branch, searching for leaves to eat. beyond among between along