Question 1 Are the associative properties true for all integers? Explain. Answer 1 The associative property states that the way numbers are grouped in addition or multiplication does not affect the result. For integers, addition and multiplication are associative, but subtraction and division are not. For example, in addition: ( 2 + 3 ) + 4 = 2 + ( 3 + 4 ) = 9 (2+3)+4=2+(3+4)=9 However, in subtraction, ( 5 − 2 ) − 1 ≠ 5 − ( 2 − 1 ) (5−2)−1  =5−(2−1). Conclusion: The associative property only works with addition and multiplication of integers, not subtraction or division. Result: No, associative properties are not true for all operations on integers. Question 2 Determine the lowest common multiple (LCM) of each of the following list of numbers: a) 9, 6 b) 8, 14 c) 2, 9, 54 Answer 1 a) LCM of 9 and 6 Multiples of 9: 9, 18, 27, 36, 45, ... Multiples of 6: 6, 12, 18, 24, 30, ... The lowest common multiple is 18. LCM(9, 6) = 18 b) LCM of 8 and 14 Multiples of 8: 8, 16, 24, 32, 40, 48, 56, ... Multiples of 14: 14, 28, 42, 56, ... The lowest common multiple is 56. LCM(8, 14) = 56 c) LCM of 2, 9, and 54 Multiples of 2: 2, 4, 6, 8, 10, 12, 14, ..., 54 Multiples of 9: 9, 18, 27, 36, 45, 54, ... Multiples of 54: 54, 108, ... The lowest common multiple is 54. LCM(2, 9, 54) = 54 Question 3 A softball team with 13 players shows up for a game. By choosing players from the 13 participants, how many different ways are there to distribute the 10 positions? Answer 1 We need to find the number of ways to arrange 10 players out of 13, where order matters (permutation). 𝑃 ( 13 , 10 ) = 13 ! ( 13 − 10 ) ! = 13 ! 3 ! P(13,10)= (13−10)! 13! = 3! 13! Calculating: = 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 1 = 1 , 037 , 836 , 800 = 1 13×12×11×10×9×8×7×6×5×4 =1,037,836,800 Thus, there are 1,037,836,800 ways to assign the 10 positions. Question 4 Indicate how many mole ratios a chemical reaction involving three substances can produce. Answer 1 For a chemical reaction with 3 substances, the mole ratio combinations can be found using 𝑛 ( 𝑛 − 1 ) n(n−1) where 𝑛 n is the number of substances. For 𝑛 = 3 n=3: 3 ( 3 − 1 ) = 6 3(3−1)=6 Result: 6 mole ratios can be produced. Question 5 In a poker hand consisting of 5 cards, find the probability of holding 3 aces. Answer 1 A standard deck has 4 aces and 52 cards total. To find the probability of 3 aces in a 5-card hand: Choose 3 aces from the 4 aces: ( 4 3 ) = 4 ( 3 4 )=4 Choose 2 cards from the remaining 48 cards: ( 48 2 ) = 48 × 47 2 × 1 = 1128 ( 2 48 )= 2×1 48×47 =1128 Total ways to get 3 aces: 4 × 1128 = 4512 4×1128=4512 Total 5-card hands possible from 52 cards: ( 52 5 ) = 2 , 598 , 960 ( 5 52 )=2,598,960 Probability: 𝑃 ( 3 aces ) = 4512 2 , 598 , 960 ≈ 0.00174 P(3 aces)= 2,598,960 4512 ≈0.00174 Result: The probability of holding 3 aces is 0.00174.