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Problem 1.9P: A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts... Problem 1.10P: In how many ways can 8 people be seated in a row if a. there are no restrictions on the seating... Problem 1.11P: In how many ways can 3 novels. 2 mathematics books, and 1 chemistry book be arranged on a bookshelf... Problem 1.12P: How many 3 digit numbers zyz, with x, y, z all ranging from 0 to9 have at least 2 of their digits... Problem 1.13P: How many different letter permutations, of any length, can be made using the letters M 0 T T 0. (For... Problem 1.14P: Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to... Problem 1.15P: Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take... Problem 1.16P: How many 5-card poker hands are there? Problem 1.17P: A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women... Problem 1.18P: A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How... Problem 1.19P: Seven different gifts are to be distributed among 10 children. How many distinct results are... Problem 1.20P: A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from... Problem 1.21P: From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How... Problem 1.22P: A person has 8 friends, of whom S will be invited to a party. a. How many choices are there if 2 of... Problem 1.23P: Consider the grid of points shown at the top of the next column. Suppose that, starting at the point... Problem 1.24P: In Problem 23, how many different paths are there from A to B that go through the point circled in... Problem 1.25P: A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3... Problem 1.26P: Show k=0n(nk)2k=3n Simplify k=0n(nk)xk Problem 1.27P: Expand (3x2+y)5. Problem 1.28P: The game of bridge is played by 4 players, each of w1om is dealt 13 cards. How many bridge deals are... Problem 1.29P: Expand (x1+2x2+3x3)4. Problem 1.30P: If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions... Problem 1.31P: If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each... Problem 1.32P: Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the... Problem 1.33P: Delegates from 10 countries, including Russia, France, England, and the United States, are to be... Problem 1.34P: If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How... Problem 1.35P: An elevator starts at the basement with 8 people (not including the elevator operator) and... 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Problem 1.8TE: Prove that (n+mr)=(n0)(mr)+(n1)(mr1)+...+(nr)(m0) Hint: Consider a group of n men and m women. How... Problem 1.9TE: Use Theoretical Exercise 8 I to prove that (2nn)=k=0n(nk)2 Problem 1.10TE: From a group of n people, suppose that we want to choose a committee of k,kn, one of whom is to be... Problem 1.11TE: The following identity is known as Fermats combinatorial identity:(nk)=i=kn(i1k1)nk Give a... Problem 1.12TE: Consider the following combinatorial identity: k=0nk(nk)=n2n1 a. Present a combinatorial argument... Problem 1.13TE: Show that, for n0 ,i=0n(1)i(ni)=0 Hint: Use the binomial theorem. Problem 1.14TE: From a set of n people, a committee of size j is to be chosen, and from this committee, a... Problem 1.15TE: Let Hn(n) be the number of vectors x1,...,xk for which each xi is a positive integer satisfying 1xin... Problem 1.16TE: Consider a tournament of n contestants in which the outcome is an ordering of these contestants,... 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Problem 1.4STPE: A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many... Problem 1.5STPE: In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts... Problem 1.6STPE: How many different 7-place license plates are possible mien 3 of the entries are letters and 4 are... Problem 1.7STPE: Give a combinatorial explanation of the identity(nr)=(nnr) Problem 1.8STPE: Consider n-digit numbers where each digit is one of the 10 integers 0,1, ... ,9. How many such... Problem 1.9STPE: Consider three classes, each consisting of n students. From this group of 3n students, a group of 3... Problem 1.10STPE: How many 5-digit numbers can be formed from the integers 1,2,... ,9 if no digit can appear more than... Problem 1.11STPE: From 10 married couples, we want to select a group of 6 people that is not allowed to contain a... 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![7.1.1 Verify the following equivalences for the age and the excess life in a renewal
process N(t): (Assume t> x.)
Pr{8x, y} = Pr{N(t-x) = N(t+y)}
= Pr{Wk<t-x, Wk+1 >t+y}
k=0
= [1 - F(t+y)]
t-x
ΣΤ
+Σ [1 − F(t+y−z)]dFk(z).
k=10
Carry out the evaluation when the interoccurrence times are exponentially dis-
tributed with parameter λ, so that dFk is the gamma density
xkzk-1
e
dFk (z) =
-zdz for z> 0.
(k-1)!](https://content.bartleby.com/qna-images/question/ba18de34-fc06-47a6-b1ea-c54726b84874/d0c34cbd-f883-44f0-86c7-1493527267ff/mgmv4cd_processed.jpeg)
