Problems

Steve, a student, has discovered that he lost most of his socks, and as a result, none of them match anymore. He has $4$ black right socks, $6$ blue right socks, $8$ black left socks, and $5$ blue left socks. Additionally, he has $2$ pairs of blue trousers and $3$ pairs of black trousers. Steve wants to ensure that his clothing items match in colour, so they desire to have left socks, right socks, and trousers of the same colour. How many different ways can Steve dress up given these conditions?

Determine the number of $5$-digit numbers that have only one odd digit and all other digits are even and distinct.

A group of $4$ adults and $5$ children is on a mountain hiking trip. At one point, the path becomes really narrow, and the hikers have to move in a line. They agreed that the line has to both start and end with an adult, for safety reasons. In how many ways can they arrange themselves?

A restaurant offers $5$ choices of starter, $10$ choices of the main course, and $4$ choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming that all food choices are available, how many different possible meals does the restaurant offer?

Find how many $5$-digit numbers contain only one $0$ and one $1$ in their decimal representation

Mary, who wrote an online computer game, plans to assign every user a unique password.
a) Currently, she wants to have passwords that are $8$ symbols long and are all made from lowercase letters $a,b,c$. How many different passwords can she generate?
b) Her friend suggested a modification: use 8-symbol words made from letters $a,b,c$, with exactly one capital letter, and also include one digit ($0-9$). How many times more passwords can she generate now?

How many numbers in the set $\{1,2,3,\dots ,1000\}$ can be expressed as a sum of $6$ consecutive integer numbers?

How many $7$-digit numbers, larger than $6$ million, are there such that the product of their digits is $42$?

On a sheet of paper a grid of $n$ horizontal and $n$ vertical straight lines is drawn. How many different closed $2n$-link broken lines can be drawn along the grid lines so that each broken line passes through all horizontal and all vertical straight lines? On the diagram below you can see an example of a closed broken line for $n=5$.

Find the largest number $A$ such that for each permutation of the set $\{1,2,3,\dots ,100\}$, the sum of some $10$ consecutive terms of that permutation is at least equal to $A$.