21_ (3 points) Suppose password is four characters long with the following restrictions_ It mnust contain exactly two instances of one symbol from the set {@, %, #,...

Passwords Suppose a password must have at least 8 characters, but no more than 12 characters, made up of letters (without distinction for case) and digits. If the password must contain at least one letter and at least one digit, how many passwords are possible?

Here we have to find out a seven character password by using a bow, All for birds on numbers. He'll be house seven spaces to filled. Oh, all for birds are numbers to Miko passport. InfoSpace off we can fill in 12 is because eight, all four birds are there on four numbers. Are there total, well, characters so we can fill in first space of Elvis. Then we can feel only 11 ways because you know character is not repeating so living, then in KASPER's, we can feel it in only 10 days off the dirt. Nine. This then eight Rays are there to fill the fifth space on seven ways to fill in six pace and six ways to fill in seven space, we will get 399 one six. Eijiro Iffy multi play all the Airborne members. We will get 3991680 in our second option if you feel only numbers in 1st 4 characters. If you feel only numbers in 1st 4 characters, how many combinations are passwords? Can we make? Let's see, 1st 4 art supposed to use only numbers. So here four numbers are there, so we can feel in four ways after dirt three V's here. Also, we are not repeating the number are all forward. So forced fourth place spaces for only numbers on then third space. We can feel it in two ways. On in fourth space, we can feel it in only one way. No, no muscle over. Only all forbids are there. Those are eight all four bets so we can feel eight. Raise in fifth space seven ways in six space in finally seven space weaken, fill in six with If you multiply, able all the numbers we will get. Eggs Seattle 64 Begin. Make 8064 passwords by using about all for birds on numbers. Barrett, 064

On this problem. We were told that the password for computer system consists of eight disdained alphabetic characters. We want to find the number of passwords possible that contain the letters word together. But in any order. Now this is a permutation problem, and permutation problems have this deal with factory ALS and so fat Toral's just have some background. In fact, oral is end times in minus one times in minus two on down to three times, two times one. Now this password here contains eight letters, but we want the letters W O R D be together. And so what we're going to do is treat that as one word, treat them as conjoined. You treat them as pin joint, and as a result, we have five different things that we want to rearrange. We have that W o R D a young think that is one and then we have these four other letters. So since we have five things to arrange, there are five factorial ways of arranging them now within this, there are for factory always of rearranging the letters inside of the word W O R D. On top of that, we have 26 options. For each of those letters, we have a through Z, so 26 options for each of them we multiplied by 26 to the fourth power. So five factorial for the number of ways of arranging the spaces for fat total for the number of ways of arranging the letters in the word word on the 26th of the fourth, the different options that we have with the other letters, and so this is the number of different arrangements that we can have.

So if we're looking at all the possible combinations in a seven character password we got 26 letters to choose from and 10 numbers to choose from and seven places to put them. So we've got 36 different characters um place in seven different positions. 2036 times 36 6 times over. Or a 36 to the power of seven. So that's gonna be all our total hub combinations Now for if we have one number and six letters that means this x right here can be the number that we have and they can go in any one of these over here. So there's seven places for it to be. There's 10 numbers to choose from and then the remainder are gonna be filled with letters. So it's gonna be 26 letters to choose from To the power of the seven minus the one which was the number that we chose to go somewhere here. So that's gonna get us 70 times 26 to the power of seven number of possible combinations. And then for a seed we are going to be looking at. And so we have 26 letters to choose from. five of them are gonna be letters And then we have 10 numbers to choose from. For the other two positions. Just going to be 100 Times 26. The power of five and the one word that makes be different from C. Is that we were looking for exactly one and this one is just the first five or letters and then the next to our numbers. And it doesn't matter which way they are since we got only two spots to put them.

Hi this is number 27 from competition combination. And the question says that there are there is a three digit password and it is known that a digit can have four values. So which means 567 or eight. It can be repeated number one. If there is exactly one correct Pastor. How many distinct wrong password? So now our objective is to find not the possible number of password but the wrong passwords. And we know only when Pastor can be right as stated. So in order to visualize the question. Yeah 1, 2 and three. So out of four years the first uh, out of four available to get my first particular password has four possibilities. Similarly the second, it will also have four possibilities and the third visit will also have four possibilities Which means it is simply for Cuba is 64 Here you will see the examiner has given a 64 other answer. But wait, this is wrong because it asked us how many distinct wrong passwords because one is correct. So we will subtract the correct path with it, which is 64 -1, which is simply 63, and therefore the correct answer will be one. I hope this explanation is clear. Uh, it was helpful. Thank you for your time.