In regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample point the sample space S for the number of possible orders at the conclusion of the contest for: - All 8 finalist - The fist 3 positions

Answer:i) There are 40320 possible ordersii) There are 336 possible orders for the first 3 positions.Step-by-step explanation:Given: The number of finalists = 8The number of boys = 3The number of girls = 5To find the number of sample point the sample space S for the number of possible orders, we need to find factorial of 8!The number of possible orders = 8!= 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8= 40320ii) From all 8 finalist, we need to choose first 3 position. Here the order is important. So we use permutation.nPr =[tex]\frac{n!}{(n - r)!}[/tex]Here n = 8 and r = 3Plug in n =8 and r = 3 in the above formula, we get8P3 = [tex]\frac{8!}{(8 - 3)!}[/tex]= [tex]\frac{8!}{5!} \\= \frac{1.2.3.4.5.6.7.8}{1.2.3.4.5}[/tex]= 6.7.8= 336So there are 336 possible orders for the first 3 positions.