{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Counting_and_Probability_Distributions_stud

# Counting_and_Probability_Distributions_stud -...

This preview shows pages 1–3. Sign up to view the full content.

Counting_and_Probability_Distributions_stud.doc Feb 24’10 PERMUTATIONS AND COMBINATIONS (handout #6; 03/15/09) THE COUNTING (MULTIPLICATION) PRINCIPLE . If one activity can occur in any of m ways, and, following this , a second activity can occur in any of n ways, then both activities can occur in the order given in m*n ways. Example . Let us consider a bank containing 4 coins: a quarter ( Q ), a dime ( D ), a nickel ( N ), and a penny ( P ). If one coin is taken out of the bank at random (the first activity), then the set of 4 possible outcomes (sample space) is {Q, D, N, P } . If a coin is tossed (the second activity), there are 2 possible outcomes, heads ( h ) and tails ( t ), and the sample space is {h, t} . If a coin is taken out of the bank and tossed, then the sample space is the set of 4*2 = 8 possible outcomes: { (Q, h), (Q, t); (D, h), (D, t); (N, h), (N, t); (P, h), (P, t) } Problem 1 . A bank contains 4 different coins: a quarter ( Q ), a dime ( D ), a nickel ( N ), and a penny ( P ), which are to be drawn out one at a time without replacement . In how many different orders can the 4 coins be removed from the bank? Solution . There are 4 possible outcomes for the first draw: Q, D, N, P. --------------------------------------------------------- Then, since 3 coins remain in the bank, | Number of Coins | Number of | there are 3 possible coins to be drawn on Draw | in the Bank | Possible Outcomes| the 2 nd draw, and in accordance with the ---------------------------------------------------------| counting principle there are 4*3 = 12 1 st | 4 | 4 | possible orders in which the first two 2 nd | 3 | 4*3 | coins could be removed: 3 rd | 2 | 4*3*2 | (Q, D), (Q, N), (Q, P); (D, Q), (D, N), (D, P); 4 th | 1 | 4*3*2*1 | (N, Q), (N, D), (N, P); (P, Q) , (P, D), (P, N). On the 3 rd draw there are 2 coins left to be selected and 4*3*2 = 24 possible orders in which the first three coins could be removed: (Q, D, N), (Q, D, P); (Q, N, D), (Q, N, P); (Q, P, D), (Q, P, N); (D, Q, N), (D, Q, P); (D, N, Q), (D, N, P); (D, P, Q), (D, P, N); (N, Q, D), (N, Q, P); (N, D, Q), (N, D, P); (N, P, Q), (N, P, D); (P, Q, D) , (P, Q, N); (P, D, Q), (P, D, N); (P, N, Q), (P, N, D). On the last draw there is one coin left to be selected and 4*3*2*1 = 24 possible orders in which the coins could have been drawn. GENERAL COUNTING (MULTIPLICATION) PRINCIPLE . If an activity 1 can occur in any of n 1 ways, and, following this , an activity 2 can occur in any of n 2 ways, an activity 3 can occur in any of n 3 ways, and so on, then all these activities can occur in n 1 * n 2 * n 3 different ways. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
DEFINITION . A permutation is an arrangement of objects in some specific order . In general, the number of permutations of n things (elements) taken r at a time n P r = n * (n – 1) * (n – 2) * …* (n – r + 1) = n!/(n – r)! , (r ≤ n) (6.1) \_________ r factors _________/ In (6.1) n! = 1 * 2 * 3 * * n means the factorial of n ; by definition of the factorial , 0! = 1 . For numerical evaluations of the number of permutations n P r the following representation of n! may be useful: n! = 1 * 2 * 3 * * n = (n – 1)! * n = (n – 2)! * (n – 1)*n = (n – 3)! * (n – 2) * (n – 1)*n (6.1a) and so on. ( CAUTION . The letter P represents here permutations , not probability .) In the special case, when r = n , the number of permutations of n things taken n at a time n P n = n * (n – 1) * (n – 2) * * 1 = n! (6.2) In terms of permutations, the number of orders in which 2 coins can be drawn out of the bank containing 4 different coins in our problem 1 is 4 P 2 = 4*3 = 12 [see (6.1)]. {The symbol 4 P 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 12

Counting_and_Probability_Distributions_stud -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online