Shmerlin managed to enter the cave and explore it. On his way back, he was once again stopped by Drago. He learns that the door out of the cave is locked again, this time with a more powerful lock. The key required to open it now includes four positive integers, which are no longer digits – they can be much larger. Shmerlin once again can choose four integer numbers: \(x, y, z\) and \(w\), and the dragon will tell him the value of \(A \times x + B \times y + C \times z + D \times w\), where \(A, B, C\) and \(D\) are the four secret integer numbers that open the lock. Because the lock is much more difficult to crack now, Drago agrees to let Shmerlin try twice. He can choose his four integer numbers and then, basing on what he learns from the dragon, choose again. Will he be able to leave the cave or is he doomed to stay inside forever?
Integer numbers \(a,b\) and \(c\) are such that the sum of digits of a number \(a+b\) is less than \(5\), the sum of digits of a number \(b+c\) is less than \(5\), the sum of digits of a number \(a+c\) is less than \(5\), but the sum of digits of a number \(a+b+c\) is greater than \(50\). Can you find such three numbers \(a,b\) and \(c\)?
Lady X has 3 different black skirts, and 5 different jackets – 3 blue, and 2 green. She also has 10 different hats – 6 blue and 4 green. Lady X’s outfit consists of a skirt, a jacket, and a hat of the matching colour.
In how many ways can the Lady choose her outfit?
Let us call a number super-odd if it is made of odd digits only. (For example, numbers \(5\), \(33\), \(13573\) are all super-odd.) How many \(3\)-digit super-odd numbers with all digits different are there?
Among 7 girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all 7 girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other?
Gabby the Gnome has 3 cloaks of different colors: blue, green, and brown. He also has 5 different hats: 3 yellow and 2 red. Finally, he owns 6 different pairs of shoes: 2 yellow, and 4 red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the color of his shoes to match the color of the hat?
A palindromic number is a number that reads the same backward and forward. (For example, \(13531\) is palindromic.)
a) How many \(5\)-digit numbers are palindromic?
b) How many \(5\)-digit numbers are palindromic and consist of distinct digits?
c) How many \(5\)-digit numbers consist of \(3\) distinct digits and end with 2?
d) How many \(5\)-digit numbers are odd and consist of \(3\) distinct digits?
\(10\) people including Alice, Bob and Charlie are waiting in a queue. How many distinct line-ups are there such that none of the mentioned three are next in the queue to both of the other two?
Mollie’s mum would like to buy 16 balloons. The balloons come in three colours: red, green, and blue. In how many ways can she buy these balloons if she would like to get at least 4 of every colour?
Mr. and Mrs. Jones have six kids – 3 boys and 3 girls. Today, a photographer is taking pictures of the family.
a) In how many ways can the kids be seated in a row so that all the girls are on the left and all the boys are on the right?
b) In how many ways can Jones’ kids be seated in a row so that girls and boys alternate?
c) In how many ways can the whole family be seated, if all the girls must be sitting together, all the boys must be sitting together as well, and parents must be either together in the centre, or on both sides?