Basic Counting Techniques: Level 2

The digital sum of a number is the sum of its digits. For how many of the positive integers 24—125 inclusive is the digital sum a multiple of 7?

Mr. Smith brings home 7 animals for his 7 children. Each child will adopt a pet to be her or his own. There are 4 different cats (a Siamese, a Persian, a Calico, and a Manx), 2 different dogs (a Poodle and a Golden Retriever), and a goldfish. Anna and Betty refuse to take care of the goldfish, and Charlie and Danny insist on having cats. The other 3 kids are easier to please–they’ll take anything. In how many ways can Mr. Smith give the children pets?

The  members of a committee are numbered 1 through . One of the members is designated as the “Grand Pooh-Bah.” The  members sit in a row of  chairs, but no member with a number greater than the Grand Pooh-Bah may sit in the seat to the immediate right of the Grand Pooh-Bah. Suppose that the Grand Pooh-Bah is member number , where . Find a formula, in terms of  and , for the number of ways for the committee to sit.

In how many ways can you spell the word NOON in the grid below? You can start on any letter, then on each step you can move one letter in any direction (up, down, left, right, or diagonal). You cannot visit the same letter twice.

A palindrome is a number that reads the same forwards and backwards, such as 12321.

(a) How many 4-digit palindromes are there?

(b) How many 5-digit palindromes are there?

(c) How many 6-digit palindromes are there?

(d) Suppose we form all -digit palindromes that consist only of 8’s and 9’s such that there is at least one of each digit. What is the smallest value of  such that there are at least 2004 numbers in the list?

Let  be a cube. Compute the number of planes which pass through at least three vertices of .

How many zeroes do we write when we write all the integers from 1 to 256 in binary?

How many 5-letter ‘words’ (where any string of letters is a ‘word’) have at least two consecutive letters which are the same?

The 10 students in my physics class sit at a round table in the following manner: The students line up in alphabetical order. The student whose name is first sits anywhere she wants. Each subsequent student chooses a chair that is next to a student who is already seated. How many different seatings are possible, assuming that two seatings are the same if each student in both seatings has the same student to the left and the same student to the right?

I have a bag with marbles numbered from 1 to 6. Mathew has a bag with marbles numbered from 1 to 12. Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles equals the number on his?