a) Establish that nCr = n! / r!(n-r)! (b) Show that 8C5 = 8C3 (c) Find the values of (i) 14!/(10!x4!) (ii) 9!/(6!x3!) (iii) 20!/(10!x10!)

a) To establish the formula for nC_r: Step 1: Define combinations. A combination nC_r represents the number of ways to choose r items from a set of n distinct items, where the order of selection does not matter. Step 2: Relate to permutations. The number of permutations of r items chosen from n is given by nP_r = (n!)/((n-r)!). Since the order of the r chosen items does not matter in a combination, we divide the number of permutations by the number of ways to arrange the r items, which is r!. Step 3: State the formula. Therefore, the formula for nC_r is: nC_r = (nP_r)/(r!) = (n!)/(r!(n-r)!) b) To show that 8C_5 = 8C_3: Step 1: Calculate 8C_5 using the formula nC_r = (n!)/(r!(n-r)!). 8C_5 = (8!)/(5!(8-5)!) = (8!)/(5!3!) Step 2: Calculate 8C_3 using the formula. 8C_3 = (8!)/(3!(8-3)!) = (8!)/(3!5!) Step 3: Compare the results. Since 5!3! = 3!5!, we can see that: (8!)/(5!3!) = (8!)/(3!5!) Thus, 8C_5 = 8C_3. c) To find the values of the expressions: i) (14!)/(10! × 4!) Step 1: Expand the factorials and simplify. (14!)/(10! × 4!) = (14 × 13 × 12 × 11 × 10!)/(10! × (4 × 3 × 2 × 1)) Step 2: Cancel 10! and calculate the remaining terms. = (14 × 13 × 12 × 11)/(4 × 3 × 2 × 1) = (14 × 13 × 12 × 11)/(24) Step 3: Perform the multiplication and division. = (24024)/(24) = 1001 The value is 1001. ii) (9!)/(6! × 3!) Step 1: Expand the factorials and simplify. (9!)/(6! × 3!) = (9 × 8 × 7 × 6!)/(6! × (3 × 2 × 1)) Step 2: Cancel 6! and calculate the remaining terms. = (9 × 8 × 7)/(3 × 2 × 1) = (9 × 8 × 7)/(6) Step 3: Perform the multiplication and division. = (504)/(6) = 84 The value is 84. iii) (20!)/(10! × 10!) Step 1: Expand the factorials and simplify. (20!)/(10! × 10!) = (20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10!)/(10! × (10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)) Step 2: Cancel 10! and simplify the remaining terms by cancelling common factors. = (20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11)/(10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) = (20/(10 × 2)) × (18/9) × (16/8) × (15/(5 × 3)) × (14/7) × (12/6) × 19 × 17 × 13 × 11 × (1/4) = 1 × 2 × 2 × 1 × 2 × 2 × 19 × 17 × 13 × 11 × (1/4) = (16 × 19 × 17 × 13 × 11)/(4) Step 3: Perform the multiplication and division. = 4 × 19 × 17 × 13 × 11 = 4 × 323 × 143 = 4 × 46189 = 184756 The value is 184756. Send me the next one 📸