2 how many different signals each consisting of eight

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2. How many different signals each consisting of eight flags hung one above the other , can be formed from a set of three indistinguishable red flags, two indistinguishable blue flags, two indistinguishable white flags and one black flag?
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COUNTING Definition(Combination) Let S be a set containing n elements and suppose r is a positive integer such that r < n . Then a combination of r elements of S is a subset of S containing r distinct elements.
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COUNTING Theorem The number of combinations of n elements taken r at a time is given by n C r = n P r / r! = n! /(n-r)! r!
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COUNTING Example 1. A football conference consists of eight teams. If each team plays every other team, how many conference games are played? 2. A student has ten posters to pin up on the walls of her room, but there is space for only seven. In how many ways can the student choose the posters to be pinned up?
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COUNTING 3. In how many ways can we select a committee of two women and three men from a group of 5 distinct women and 6 distinct men? 4. How many poker hands contain cards all of the same suit? 5. From 6 history books and 8 language books, in how many ways can a person select two history books and three language books and arrange them on a shelf?
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COUNTING Binomial Theorem If n is any positive integer, then ( 29 n n n r r n r n n n n n n n n b C ... b a C ... b a C b a C a C b a + + + + + + = + - - - 2 2 2 1 1 0
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PROBABILITY PROBABILITY Random experiment Sample space Event as a subset of sample space Likelihood of an event to occur- probability of an event
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PROBABILITY Features of a Random Experiment all outcomes are known in advance. The outcome of any one trial cannot be predicted with certainty. Trials can be repeated under identical conditions.
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PROBABILITY Examples (Random Experiment) Rolling a die and observing the number of dots on the upturned face. Tossing a coin and observing the upturned face.
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PROBABILITY Sample Space It is a set such that each element denotes an outcome of a random experiment. It is usually denoted by or S. Example . Rolling a die and observing the number of dots on the upturned face. S = {1,2,3,4,5,6}
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PROBABILITY Event A subset of the sample space whose probability is defined. Example S = {1,2,3,4,5,6} a. An event of observing odd number of dots in a roll of a die : E 1 = {1,3,5} b. An event of observing even number of dots in a roll of a die : E 2 = {2,4,6}
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PROBABILITY Two events are mutually exclusive if the two events cannot occur simultaneously. Example Coin toss: either a head or a tail, but not both. The events head and tail are mutually exclusive.
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PROBABILITY Definition(Probability) The probability of an event E , which is a subset of a finite sample space S of equally likely outcomes is P(E) = |E|/|S| note: A probability function P assigns to each outcome x in a sample space S a number P(x) so that 0 < P(x) < 1 for all x in S and P(x) = 1 .
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PROBABILITY Example: If we select a card at random from a well shuffled deck of cards then, P(ace in a deck of cards)= 4/52 Note: Probability of a joint event, A and B P(A and B) = P(A∩B) Example P(red card and Ace) = 1/26
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PROBABILITY
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