Strings, Sets, and Binomial Coefficients

Matt is designing a website authentication system. He knows passwords are most secure if they contain letters, numbers, and symbols. However, he doesn't quite understand that this additional security is defeated if he specifies in which positions each character type appears. He decides that valid passwords for his system will begin with three letters (uppercase and lowercase both allowed), followe…

How many ternary strings of length $2n$ are there in which the zeroes appear only in odd-numbered positions?

Suppose we are making license plates of the form $l_1l_2l_3-d_1d_2d_3$ where $l_1,l_2,l_3$ are capital letters in the English alphabet and $d_1,d_2,d_3$ are decimal digits (i.e., elements of the set $\{0,1,2,3,4,5,6,7,8,9\}$) subject to the restriction that at least one digit is nonzero and at least one letter is $K$. How many license plates can we make?

Mrs. Steffen's third grade class has $30$ students in it. The students are divided into three groups (numbered $1$, $2$, and $3$), each having $10$ students.

1. The students in group $1$ earned $10$ extra minutes of recess by winning a class competition. Before going out for their extra recess time, they form a single file line. In how many ways can they line up?

2. When all $30$ students come in from recess together, they a…

In this exercise, we consider strings made from uppercase letters in the English alphabet and decimal digits. How many strings of length $10$ can be constructed in each of the following scenarios?

1. The first and last characters of the string are letters.

2. The first character is a vowel, the second character is a consonant, and the last character is a digit.

3. Vowels (not necessarily distinct) appear in the…

A database uses $20$-character strings as record identifiers. The valid characters in these strings are upper-case letters in the English alphabet and decimal digits. (Recall there are $26$ letters in the English alphabet and $10$ decimal digits.) How many valid record identifiers are possible if a valid record identifier must meet all of the following criteria:

• Letter(s) from the set $\{A,E,I,O,U\}$ occur in exactly thr…

Let $X$ be the set of the $26$ lowercase English letters and $10$ decimal digits. How many $X$-strings of length $15$ satisfy all of the following properties (at the same time)?

• The first and last symbols of the string are distinct digits (which may appear elsewhere in the string).

• Precisely four of the symbols in the string are the letter '$t$'.

• Precisely three characters in the string are elements of the set $V=\{a,e,i,o,u\}$ an…

The sport of korfball is played by teams of eight players. Each team has four men and four women on it. Halliday High School has seven men and $11$ women interested in playing korfball. In how many ways can they form a korfball team from their 18 interested students?

An ice cream shop has a special on banana splits, and Xing is taking advantage of it. He's astounded at all the options he has in constructing his banana split:

• He must choose three different flavors of ice cream to place in the asymmetric bowl the banana split is served in. The shop has 20 flavors of ice cream available.

• Each scoop of ice cream must be topped by a sauce, chosen from six different o…

Suppose that a teacher wishes to distribute $25$ identical pencils to Ahmed, Barbara, Casper, and Dieter such that Ahmed and Dieter receive at least one pencil each, Casper receives no more than five pencils, and Barbara receives at least four pencils. In how many ways can such a distribution be made?

How many integer solutions are there to the equation provided that $x_1>0$, and $x_2,x_3,x_4\geq 0$? What if we add the restriction that $x_4\lt 17$?

How many integer solutions are there to the inequality provided that $x_1,x_2>0$, $x_3\geq 0$, and $x_4,x_5\geq 10$?

A teacher has $450$ identical pieces of candy. He wants to distribute them to his class of $65$ students, although he is willing to take some leftover candy home. (He does not insist on taking any candy home, however.) The student who won a contest in the last class is to receive at least $10$ pieces of candy as a reward. Of the remaining students, $34$ of them insist on receiving at least one piece of candy,…

Let $m$ and $w$ be positive integers. Give a combinatorial argument to prove that for integers $k\geq 0$,

How many lattice paths from $(0,0)$ to $(14,73)$ are there that do not pass through $(6,37)$?

The setting for this problem is the fictional town of Mascotville, which is laid out as a grid. Mascots are allowed to travel only on the streets, and not “as the yellow jacket flies.” Buzz, the Georgia Tech mascot, wants to go visit his friend Thundar, the North Dakota State University mascot, who lives $6$ blocks east and $7$ blocks north of Buzz's hive. However, Uga VIII has recently moved into the…

Determine the coefficient on $x^{12}y^{24}$ in $(x^3+2xy^2+y+3)^{18}$. (Be careful, as $x$ and $y$ now appear in multiple terms!)