Question

1. you will make a five character strings from the letters {a,e,i,o,u} and {1,2,3,4,5,6}.

(a) if characters can be used any number of times how many different strings can be made in this way?
(b) if characters can be used at most once how many different strings can be made in this way?
(c) if characters are allowed to be repeated how many strings are there such that at least one character appears more than once in the string?
(d) if characters cannot be repeated how many strings are there which alternate between letters and numbers, and a letter appears first?

Explanation:

(a)

Step1: Determine the number of choices for each character

There are \(5 + 6=11\) characters in total (\(5\) vowels and \(6\) numbers). Since each of the \(5\) positions in the string can be filled with any of the \(11\) characters (repetition allowed), we use the multiplication principle.
The number of ways to form the string is \(N_1=11\times11\times11\times11\times11 = 11^{5}\)

$$11^{5}=161051$$

(b)

Step1: Use the permutation formula

The total number of characters is \(n = 5+6=11\). We want to arrange \(r = 5\) non - repeating characters. The permutation formula is \(P(n,r)=\frac{n!}{(n - r)!}\)

$$P(11,5)=\frac{11!}{(11 - 5)!}=\frac{11!}{6!}=11\times10\times9\times8\times7=55440$$

(c)

Step1: Use the principle of complementary counting

The total number of strings with repetition allowed is \(T=11^{5}\) (from part (a)). The number of strings with no repetition is \(N = P(11,5)\) (from part (b)).
The number of strings with at least one repetition is \(11^{5}-P(11,5)\)

$$11^{5}-P(11,5)=161051 - 55440=105611$$

(d)

Step1: Analyze the pattern of the string

Since the string alternates between letters and numbers and starts with a letter, the pattern is \(L - N - L - N - L\) (where \(L\) represents a letter and \(N\) represents a number)

Step2: Calculate the number of choices for each position

For the first (letter) position, there are \(5\) choices. For the second (number) position, there are \(6\) choices. For the third (letter) position, there are \(4\) choices (since we cannot repeat the first letter). For the fourth (number) position, there are \(5\) choices (since we cannot repeat the second number). For the fifth (letter) position, there are \(3\) choices (since we cannot repeat the first and third letters)
The number of such strings is \(N=5\times6\times4\times5\times3\)

$$5\times6\times4\times5\times3 = 1800$$

Answer:

(a) \(161051\)
(b) \(55440\)
(c) \(105611\)
(d) \(1800\)