# A Calculus for Factorial Arrangements (Lecture Notes in Statistics)

How many of these outcomes have 6 tails? When you flip a coin once, there are two possible outcomes; a head or a tail. If you flip the coin more than once, the out comes appear in combinations of heads and tails: In other words, we're looking for combinations!

Therefore the question is asking for. This question is really simple, the trick is to ignore that misleading choice of word combinations. This question is about permutations since we've been asked to arrange the letters without any order in mind. The only difference here is that we have been asked that the first 3 letters of all the different permutations must be 'BAN'. The solution is to subtract the number of letters whose position is constant and then permutate the remaining letters:. Factorials, Permutations and Combinations Factorials A factorial is represented by the sign!

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This works out to be mathematically true and allows us to redefine n! For example The above allows us to manipulate factorials and break them up, which is useful in combinations and permutations. Useful Factorial Properties The last two properties are important to remember. Permutations and Combinations Permutations and Combinations in mathematics both refer to different ways of arranging a given set of variables. For example; given the letters abc The Permutations are listed as follows Combinations on the other hand are considered different, all the above are considered the same since they have the exact same letters only arranged different.

In groups of 1 we get In groups of 2 we get In groups of 3 we get From the above, you should see that Combinations are about finding how many ways you can combine the different elements of the given entity. The notation for Combinations is given as which means the number of combinations of n items taking r items at a time For example means find the number of ways 3 items can be combined, taking 2 at a time, and from the example before, we saw that this was 3. Another example to further illustrate this is as follows: Given four letters abcd find solution: The Combination function can be defined using factorials as follows: We can prove that this is true using the previous example; which is the same answer we got before.

Permutations are denoted by the following which means the number of permutations of n items taken r items at a time. For example; given 3 letters abc find solution: Remember that the repetition is allowed in permutations unlike in combinations; which mean that there are 6 ways, in other words The Permutation function can also be using factorials: We can prove the above using the previous example Which is the same answer as before.

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The above can be proved by substituting the formula for permutations into the equation Which as we already saw is the formula for Combinations. Examples of Factorials, Permutations and Combinations Example 1 Evaluate the following without using a calculator. Step 1 We have seen that a relatively big number like 10 in this example can be broken down into a product of factorials i. Step 2 We can use the above to evaluate the expression as.

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Step 3 Since 7! Example 2 Evaluate the following. Step 1 We have already defined the combination notation above as: Step 2 Therefore, we can just substitute in the above formula. Step 3 The numerator and denominator are equal so we can just cancel them out as. Example 3 Evaluate the following expression. Step 1 The notation above shouldn't be all that unfamiliar if you've gone through the page this entire page. Special offers and product promotions Rs cashback on Rs or more for purchases made through Amazon Assistant.

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## A Calculus for Factorial Arrangements - Sudhir Gupta, Rahul Mukerjee - Google Книги

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