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1 Let X be the set of all different eight letter words that can be obtained by permuting the letters of the word DEBUNKED How many elements does X contain In how many of them there are neighbouring identical letters In how many elements of X there are no neighbouring vowels terminology E U are vowel letters while B D K N are consonant letters 2 Assume that a basketball line up must consist of 5 players 2 guards 1 center and 2 forwards A coach has in total 12 players in the team there are 5 guards 2 centers 4 forwards and additionally Peter Williams who can play as center or forward How many 5 player subsets of this 12 element set are acceptable line ups In how many of them does Peter Williams appear 3 In how many ways can we put 20 identical silver coins into five coloured boxes so that at most 3 coins go into the blue box at least 4 into red and at most 5 into green The remaining boxes are yellow and black and may contain any number of coins Every box except the red one may also remain empty 4 How many solutions of the equation a b c d 80 in integers greater than 0 satisfy simultaneously all the following conditions a 30 10 c 40 and a b c d are all even REMARK If your answer is eg 3 56 you don t have to compute the numerical answer 546875 Just write 3 56 as the final answer

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