(4) Let S = {a,b,c,d,e}. (a) List all of the 2-permutations of S in an organized manner. How many are there? (b) List all of the 2-combinations of S in an organized manner. How many are there? (c) List all of the 3-combinations of S in an organized manner. How many are there? (d) On the previous page, you should have gotten the same number of 2-combinations of S as 3-combinations. Explain why we should expect these numbers to be the same.

Answer with Step-by-step explanation:We are given that a set S={a,b,c,d,e}a.We have n=5We have to find the number of 2-permutations of S in an organised mannerUsing permutation formula [tex]nP_r=\frac{n!}{r!(n-r)!}[/tex]r=2 [tex]5P_2=\frac{5!}{3!}[/tex][tex]5P_2=\frac{5\times 4\times3!}{3!}[/tex][tex]5P_2=20[/tex]Hence, the total number 2- permutations of S =20{a,b},{b,c},{c,d},{d,e},{a,c},{a,d},{a,e},{b,d},{b,e},{c,e},{b,a},{c,b},{d,c},{e,d},{c,a},{d,a},{e,a},{d,b},{e,b},{e,c}.b.We have to find the 2- combinations of S in an organised manner By using combination formula[tex]nC_r=\frac{n!}{r!(n-r)!}[/tex]n=5,r=2[tex]5C_2=\frac{5!}{2!(5-2)!}[/tex][tex]5C_2=\frac{5\times 4\times3!}{2!3!}[/tex][tex]5C_2=10[/tex]Hence, the number of 2 -combinations of S =10{a,b},{b,c},{c,d},{d,e},{a,c},{a,d},{a,e},{b,d},{b,e},{c,e}.c.We have n=5 r=3Using combination formula[tex]5C_3=\frac{5!}{3!(5-3)!}[/tex][tex]5C_3=\frac{5\times 4\times 3!}{3!2!}[/tex][tex]5C_3=10[/tex]Hence, the total number of 3- combinations of S =10{a,b,c},{b,c,d},{c,d,e},{a,b,d},{a,b,e},{b,c,e},{a,c,d},{a,d,e},{b,d,e},{a,c,e}.d.Number of 2 -combinations of S=Number of 3- combinations of S=10Combinations is a selection of r elements out of n elements.When we select 3 elements  out of 5 then we get total number of combinations of S=10 and when we select 2 elements out of 5 then we get total number of combinations of S=10By combinations formula [tex]5C_3=10[/tex][tex]5C_2=10[/tex]