a) i) If 4 individuals are selected from a group of 7 people, how many possible selection option could be made?ii) In a competition prizes are given to 6 of the 10 participants. How many possible prize lists could be made if they are given different prizes?b) In how many ways can the letters of the word SEMESTER be rearranged?

Answer:(a) 35 options(b) 151,200 permutations(c) 3,360 waysStep-by-step explanation:a) The number of combinations of p=4 people out of a n=7 can be calculated as[tex]C=\frac{n!}{p!(n-p)!}=\frac{7!}{4!*3!}  =\frac{5040}{24*6}=35[/tex]b) In this case is a permutation with no repetition of r=6 elements in a n=10 elements group. [tex]P=\frac{n!}{(n-r)!}=\frac{10!}{(10-6)!}=\frac{3628800}{24}=151,200[/tex]c) The word semester has 2 S's, 3 E's, 1 M, 1 T and 1 R (8 letters in total).The total amount of permutations can be calculated as the total amount of permutations of letters, divided by the factorial of the amount of times a same letter repeats in the word. Or is the same to say, dividing by the amount of repetitions of the same permutation. [tex]P=\frac{8!}{3!2!} =\frac{40320}{6*2}=3,360[/tex]