Question

5.1 Combination and permutation

a) ₅C₂

b) 5!

c) Five different drugs, A, B, C, D, and E, can be used to treat a disease in different combinations. If a physician uses two of them to treat patients, how many combinations are possible? List all such combinations.

d) Four different exercises, A, B, C, and D, are recommended for an injury recover therapy program. The therapists want to know whether there is a different treatment effect using i) just one; ii) any two; or iii) any three of the therapies. How many treatment regimens can a therapist choose from? What are they?

e) If the order matters in d), if therapists were to select two exercises out of four, how many treatment options would the therapists have? What are they?

Answer 1

a) ⁵C₂ = 5! / (2! * 3!) = 10

b) 5! = 5*4*3*2*1 = 120

c) Number of combination to treat disease = ⁵C₂ = 5! / (2! * 3!) = 10

List has 10 entries.
{A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, {C,E}, {D,E}

d) For just one:

Number of combination are : ⁴C₁ = 4! / (1! * 3!) = 4

For any two:

Number of combination are : ⁴C₂ = 4! / (2! * 2!) = 6

For any three:

Number of combination are : ⁴C₃ = 4! / (3! * 1!) = 4

Total Number of treatment regimens a therapist can choose from = 4+6+4 = 14

List:

{A} {B} {C} {D}

{A,B} {A,C} {A,D} {B,C} {B,D} {C,D}

{A,B,C} {A,B,D} {A,C,D} {B,C,D}

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e) For just one:

Number of combination are : ⁴P₁ = 4! / ( 3!) = 4

For any two:

Number of combination are : ⁴P₂ = 4! / (2!) = 12

For any three:

Number of combination are : ⁴P₃ = 4! / (1!) = 24

Total Number of treatment regimens a therapist can choose from = 4+12+24= 40

List:

{A} {B} {C} {D}

{A,B} {A,C} {A,D} {B,A} {B,C} {B,D} {C,A} {C,B} {C,D} {D,A} {D,B} {D,C}

{A,B,C} {A,B,D} {A,C,B} {A,C,D} {A,D,B} {A,D,C} {B,A,C} {B,A,D} {B,C,A} {B,C,D} {B,D,A} {B,D,C} {C,A,B} {C,A,D} {C,B,A} {C,B,D} {C,D,A} {C,D,B} {D,A,B} {D,A,C} {D,B,A} {D,B,C} {D,C,A} {D,C,B}

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