Suppose we wish to create a banner containing a string of letters n inches long. The letters we wish to use are available in the following widths: 1 inch wide letters: f, i, t . 2 inch wide letters: a, c, d, e, g, n, o, p, s, u For example, the string "stopsign" is 14 inches long. a. How many different ways can we create a string whose length is 0 inches? How many different ways can we create a string whose length (in inches) is 1? 2? 3? 4? For full credit, make sure it is clear how you came up with your results (i.e., just giving the numeric values will not be sufficient). Note that instances of the same letter are indistinguishable and the order of the letters in the string does matter. For example, there is only one way to have a string with 3 t's (since t's are not distinguished) and there are two ways to have a string with one c and one s: "cs" and "sc". Note also that the strings are not required to be english words and don't contain any spaces. b. Find a recurrence relation for the number of ways a string of letters n inches long can be created. Make sure to justify your answer. Suppose we wish to create a banner containing a string of letters n inches long. The letters we wish to use are available in the following widths: 1 inch wide letters: f, i, t . 2 inch wide letters: a, c, d, e, g, n, o, p, s, u For example, the string "stopsign" is 14 inches long. a. How many different ways can we create a string whose length is 0 inches? How many different ways can we create a string whose length (in inches) is 1? 2? 3? 4? For full credit, make sure it is clear how you came up with your results (i.e., just giving the numeric values will not be sufficient). Note that instances of the same letter are indistinguishable and the order of the letters in the string does matter. For example, there is only one way to have a string with 3 t's (since t's are not distinguished) and there are two ways to have a string with one c and one s: "cs" and "sc". Note also that the strings are not required to be english words and don't contain any spaces. b. Find a recurrence relation for the number of ways a string of letters n inches long can be created. Make sure to justify your