You will learn to calculate objects in simple, combinatorics situations while using the rule of sum and rule of product.
Learning effect
You calculate objects in simple, combinatorics situations while using the rule of sum and rule of product.
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Prepare information about the rule of sumrule of sumrule of sum and the rule of productrule of productrule of product and an answer for the question “what is combinatoricscombinatoricscombinatorics?”
See if in information you prepared there are following definitions:
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Combinatorics is the theory of defining number of elements of finite sets or structures of this elements.
The number of all possible outcomes of an experiment that involves doing one of n activities, out of which the first can end in one of kIndeks dolny 11 ways, the second – in one of kIndeks dolny 22 ways, the third – in one of kIndeks dolny 33 ways and so on till n‑th activity that can end in one of kIndeks dolny nn ways, is equal to kIndeks dolny 11 + kIndeks dolny 22 + kIndeks dolny 33 +…+ kIndeks dolny nn.
The number of all possible outcomes of an experiment that involves doing n activities one by one, out of which the first can end in one of kIndeks dolny 11 ways, the second – in one of kIndeks dolny 22 ways, the third – in one of kIndeks dolny 33 ways and so on till n‑th activity that can end in one of kIndeks dolny nn ways, is equal to kIndeks dolny 11· kIndeks dolny 22· kIndeks dolny 33·…· kIndeks dolny nn.
Task 1
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Open the interactive illustration that presents ways of applying the rule of sum and the rule of product.
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After having completed the exercise, present results of your observations and compare them with the following ones:
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If we make a few independent partial decisions that are included in one whole choice, then we multiply the number of decisions, but if we make mutually exclusive choices, then we add the number of choices.
Task 2
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How many ways of creating the timetable for one day there are, if in this day there should be one class of Polish, mathematics, history, biology, IT and English?
Task 3
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How many four‑digit codes can be made only from digits from the set {2,4,6,7,8}, if:
a. digits can be repeated b. each digit can be used only once.
Task 4
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In a bar, there are 5 different soups, 7 main dishes and 4 kinds of drinks. How many sets made of soup, main dish and drink can we order in this bar?
Task 5
An extra task:
How many positive integer divisors of the number 132300 are there?
The number of all possible outcomes of an experiment that involves doing one of n activities, out of which the first can end in one of kIndeks dolny 11 ways, the second – in one of kIndeks dolny 22 ways, the third – in one of kIndeks dolny 33 ways and so on till n‑th activity that can end in one of kIndeks dolny nn ways, is equal to kIndeks dolny 11 + kIndeks dolny 22 + kIndeks dolny 33 +…+ kIndeks dolny nn.
The number of all possible outcomes of an experiment that involves doing n activities one by one, out of which the first can end in one of kIndeks dolny 11 ways, the second – in one of kIndeks dolny 22 ways, the third – in one of kIndeks dolny 33 ways and so on till n‑th activity that can end in one of kIndeks dolny nn ways, is equal to kIndeks dolny 11· kIndeks dolny 22· kIndeks dolny 33·…· kIndeks dolny nn.
Exercises
Exercise 1
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Exercise 2
Wojtek has 3 jackets, 6 shirts and 5 ties. How many different sets made of a jacket, a shirt and a tie can he make?
90
Exercise 3
There are 15 girls and 8 boys in the class III a. A two‑persons delegation needs to be chosen, made of one girl and one boy. How many possibilities of such choice are there?