Understanding Permutations and Combinations in Mathematics
Explore permutations and combinations to count arrangements of objects using examples with letters and words. Learn how to calculate permutations and combinations, along with practical applications in team selection scenarios. Enhance your mathematical knowledge with clear explanations and visual aids on mathssupport.org.
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18 June 2025 Permutations LO: Use permutations to count arrangements. www.mathssupport.org
Permutations A permutation of a group of symbols is any arrangement of those symbols in a definite order. For example: Given the letters A, B and C List all the permutations when they are taken 1 at a time A, B, C AB, BA, CA AC, BC, CB ABC,BAC,CAB ACB,BCA,CBA 2 at a time 3 at a time www.mathssupport.org www.mathssupport.org
Permutations For example: Given the letters P, Q, R and S List all the permutations when they are taken P, Q, R, 1 at a time S RP, PS, SP SQ,RS, SR PQ, QP, PR, QR, RQ,QS, 2 at a time 12 Permutations The number of permutations of r objects out of n distinct objects is nPr = Pr ?! n= ? ? ! In the example 4= 4 2 ! r = 2 n = 4 2!=4 3 2! =4! 4! = 12 P2 2! www.mathssupport.org www.mathssupport.org
Permutations Example 1: In how many ways can the letters of the word example be arranged n = 7 r = 7 0!=7! =7! 7! 7= P7 = 7! = 5040 1 7 7 ! In how many ways can the letters be arranged taking them two at a time? n = 7 r = 2 =7 6 5! =7! 7! 7= P2 = 42 5! 5! 7 2 ! www.mathssupport.org
Combinations LO: Use combinations to count arrangements. www.mathssupport.org
Combinations A combination is a selection of objects without regard to order or arrangement. For example: Given the letters A, B and C List all the combinations when they are taken AB, BC, AC BA, CB, CA 2 at a time Are the same combinations Only 3 combinations www.mathssupport.org www.mathssupport.org
Combinations For example: Given the letters P, Q, R, S and T List all the combinations when they are taken 3 at a time PQR, PQS, PQT, QRS, RST, The number of combinations on n distinct objects taken r objects at a time is: In the example 5= 3! 5 3 != PRS, PRT, PST QST, QRT, 10 Combinations r = 3 n = 5 5! 3!2!=5 4 3! ?! n= nCr= Cr 5! = 10 C3 ?! ? ? ! 3!2! www.mathssupport.org www.mathssupport.org
Combinations Example 1: How many different teams of 4 can be selected from a squad of 7, if there are no restrictions? r = 4 n = 7 7! 4!3!=7 6 5 4! 7! 7= C4 = 35 4! 7 4 != What if the teams must include the captain C1 of the other 6 4!3! 6 1 And we need any 3 C3 the captain must be included =6 5 4 3! 3!(6) 1! 6! 3! 6 3 !=1! 6! 1 6 C1 C3 = 1! 1 1 ! = 20 0! 3!3! www.mathssupport.org
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