Name Surname:

Student Number:

You don’t need to find exact results. Just do the operations. You have 30 mins.

  1. In how many ways can you arrange the letters of “SERENDIPITY”?

    1. Any order?
    2. Vowels together?

Solution: SERENDIPITY has 11 letters: 4 vowels, 7 consonants. There are 2 Es and 2 Is.

  1. Remember the same letters should be accounted for. \(\binom{11!}{2!2!}\)

  2. Suppose vowels are a single word (i.e. X is any permutation of EEII), call it a special word. So a permutation can be SRNDPTYX, another would be NRSXDPTY. Also “within” X, vowels can get different permutations (e.g. EIEI or IEEI). If we replace X with the vowels some proper permutations would be SRNDPTYEIEI, SRNDPTYIEEI, NRSEIEIDPTY, NRSIEEIDPTY.

    First assume vowels do not change place. Then there are \(8!\) permutations. Now calculate vowel permutations \(\binom{4!}{2!2!}\). For each vowel permutation and a special word (i.e. SRNDPTYX) permutation, there is a valid permutation (i.e. vowels together).

    So the answer is \(8!*\binom{4!}{2!2!}\).

  1. In a box there are 12 balls, 4 white 8 black. If I randomly pick 5 balls from the box, what is the probability that it will be 1 white and 4 blacks?

Remember Hypergeometric distribution.

\[ \dfrac{\binom{4}{1}\binom{8}{4}}{\binom{12}{5}} \]

  1. John’s basketball shots score with probability 0.8. What is the probability that at least three scores among the first 10 shots?

Solution: It is a binomial distribution problem. A single probability (i.e. scoring exactly k shots) can be calculated as follows \(\binom{n}{k}p^k(1-p)^{n-k}\). There are two ways to calculate the correct answer. First one is to calculate scores from 3 to 10, second one is calculate from 0 to 2 and subtract from total probability (i.e. 1). Below expression belongs to the second solution.

\[ 1 - P(X=0) - P(X=1) - P(X=2) = 1 - (0.2)^10 - 10*(0.8)*(0.2)^9 - \binom{10}{2}*(0.8)^2*(0.2)^8 \]