Exam 6: Counting

Find the number of positive integers not exceeding 1000 that are not divisible by 4, 6, or 9.

How many subsets with an odd number of elements does a set with 10 elements have?

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have at least one C.

Aprocomputerve that someis programmedsubset musttohaprinve bteensubsetsprintedof at{1,least2, 3, 4t,wice.5} at random. If the computer prints 40 subsets,

In questions suppose you have a class with 30 students - 10 freshmen, 12 sophomores, and 8 juniors. -In how many ways can you get a committee of 7?

If the permutations of 1,2,3,4,5,6 are written in lexicographic order, with 123456 in position # 1,123465 in position #2, etc., find the permutation immediately after 246531 .

In questions let A be the set of all bit strings of length 10. -How many bit strings of length 10 have more 0's than 1's? In questions 45-48 suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all 30 books are different. In how many ways can you

Find the number of terms in the expansion of $( 5 a + 8 b ) ^ { 15 }$

You have 20 cards and 12 envelopes (labeled 1, 2, . . . , 12). In how many ways can you put the 20 cards into the envelopes if (a) the cards are distinct. (b) the cards are identical. (c) the cards are identical and no envelope can be left empty.

In questions let A be the set of all strings of decimal digits of length five. For example 00312 and 19483 are strings in -How many strings in A have exactly three $5 \mathrm {~s} ?$

Find the number of words of length eight of distinct letters of the alphabet so that the words do not have both A and B in them.

Use the binomial theorem to expand $( 2 a + b ) ^ { 4 }$

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

How many ways are there to select 6 students from a class of 25 to serve on a committee?

Make up a word problem in good English whose answer is $15 ! / 10 !$

How many different strings can be made using all the letters in the word GOOGOL?

How many permutations of the seven letters A, B, C, D, E, F, G have E in one of the first two positions?

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with a vowel or end with a vowel?

Findeven. the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, the sum of which is

Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit.