CS206 -Discrete Mathematics II
Instructor: Chitoor V.Srinivasan
SOLUTIONS TO REVIEW PROBLEM SET 3

  1. Since we have

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    And if we can show that

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    Then we can finish our proof. Now let's use mathematics induction to prove the above formula.
    Basis: tex2html_wrap94

    Induction Hypothesis: Let's assume following holds.

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    Induction Step: If we can show

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    then it's all done.

    According to Theorem 2.3.6, we have

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  2. Since  A  and  B  are events defined over  S ,hence we have   tex2html_wrap_inline142 . And

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    Substituting   tex2html_wrap_inline144 , we get

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  3. The sample outcomes which the sum of the two fair dices exceeds  8  are following

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    The sample outcomes which the sum of the two fair dices equals  10  are following

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    Hence the probability that the sum equals  10  given it exceeds  8  is  3/10.

  4. Let  B  denote that the chip drawn from urn II is red, and   tex2html_wrap_inline158   denote that the chip drawn from urn I and transferred to urn II is red, and   tex2html_wrap_inline160   denote that the chip drawn from urn I and transferred to urn II is red, hence we get

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  5. Poisson Distribution   tex2html_wrap_inline162 ,  where   tex2html_wrap_inline164   is the mean value.

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  6. The chance of a point selected randomly locates inside the triangle is the ratio of the area of triangle, which is   tex2html_wrap_inline166 , to the area of circle, which is   tex2html_wrap_inline168 . And we consider it that we choose points randomly as Bernolli Trials, hence we get the chance of a point located inside the triangle, p,  which is   tex2html_wrap_inline172 , and the chance of a point located outside the triangle, q, which is   tex2html_wrap_inline176 .

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  7. By the multinomial theorem.

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    Since the only way to use   tex2html_wrap_inline178   to construct  13  is to sum ninty-seven  0, two  4  and one  5. Thus the coefficient of   tex2html_wrap_inline188   is   tex2html_wrap_inline190 .

  8. Lets assume the length of the series of random digits is  n, and we have   tex2html_wrap_inline194   ten digits. Probability that a particular digit is neither 5 nor 7 is  8/10. Hence probability of getting at least one  5/7 is   tex2html_wrap_inline200   . Solve

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    to get the minimum value of n required.

  9. We have totally  32  cards, which are from  2  to  8. There are   tex2html_wrap_inline208   ways to draw five cards from the thirty-two cards.

  10. There are totally   tex2html_wrap_inline210   random function from the domain   tex2html_wrap_inline212   to the range   tex2html_wrap_inline214 , since each item in the domain will be able to map  m  items in the range. If the function would have exactly  k  elements in its range, it means the random function is an onto function. Besides we will have   tex2html_wrap_inline220   ways to choose the range of the onto function. For a spcific range of  k  elements, the number of onto function is

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    Hence the total number of onto function from the domain   tex2html_wrap_inline212   to the range of  k  elements is

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    Thus the probability that the onto function would have exactly  k  elements in its range is

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