91Ó°ÊÓ

a. How many distinguishable ways can the letters of the word \(H U L L A B A L O O\) be arranged? b. How many distinguishable arrangements of the letters of \(H U L L A B A L O O\) begin with \(U\) and end with \(L\) ? c. How many distinguishable arrangements of the letters of \(H U L L A B A L O O\) contain the two letters \(H U\) next to each other in order?

a. How many distinguishable ways can the letters of the word \(M I L L I M I C R O N\) be arranged? b. How many distinguishable arrangements of the letters of \(M I L L I M I C R O N\) begin with \(M\) and end with \(N\) ? c. How many distinguishable arrangements of the letters of MILLIMICRON contain the letters \(C R\) next to each other in order and also the letters \(O N\) next to each other in order?

a. How many integers are there from 1000 through 9999 ? b. How many odd integers are there from 1000 through \(9999 ?\) c. How many integers from 1000 through 9999 have distinct digits? d. How many odd integers from 1000 through 9999 have distinct digits? e. What is the probability that a randomly chosen four-digit integer has distinct digits? has distinct digits and is odd?

Consider strings of length \(n\) over the set \(\\{a, b, c, d\\}\). a. How many such strings contain at least one pair of adjacent characters that are the same? b. If a string of length ten over \([a, b, c, d]\) is chosen at random, what is the probability that it contains at least one pair of adjacent characters that are the same?

a. How many positive two-digit integers are multiples of 3 ? b. What is the probability that a randomly chosen positive two-digit integer is a multiple of 3 ?

a. How many integers from \(\mid\) through 1,000 are multiples of 4 or multiples of 7 ? b. Suppose an integer from 1 through 1,000 is chosen at random. Use the result of part (a) to find the probability that the integer is a multiple of 4 or a multiple of 7 . c. How many integers from I through 1,000 are neither multiples of 4 nor multiples of 7 ?

A fair coin is tossed until either a head comes up or four tails are obtained. What is the expected number of tosses?

Prove that if \(A\) and \(B\) are independent events in a sample space \(S\), then \(A^{c}\) and \(B^{c}\) are also independent.

a. How many integers from 1 through 1,000 are multiples of 2 or multiples of 9 ? b. Suppose an integer from I through 1,000 is chosen at random. Use the result of part (a) to find the probability that the integer is a multiple of 2 or a multiple of \(9 .\) c. How many integers from 1 through 1,000 are neither multiples of 2 nor multiples of 9 ?

Suppose \(A[1], A[2], A[3], \ldots, A[n]\) is a one-dimensional array and \(n \geq 50\). a. How many elements are in the array? b. How many elements are in the subarray $$ A[4], A[5], \ldots, A[39] \text { ? } $$ c. If \(3 \leq m \leq n\), what is the probability that a randomly chosen array element is in the subarray $$ A[3], A[4], \ldots, A[m] \text { ? } $$ d. What is the probability that a randomly chosen array element is in the subarray shown below if \(n=39\) ? $$ A[\lfloor n / 2\rfloor], A[\lfloor n / 2\rfloor+1], \ldots, A[n] $$