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How many? 7-digit telephone numbers are possible if the firstdigit cannot be two and ?(a) only even digits may be? used? ?(b)the number must be a multiple of 10? (t...

Question

How many? 7-digit telephone numbers are possible if the firstdigit cannot be two and ?(a) only even digits may be? used? ?(b)the number must be a multiple of 10? (that is, it must end in? 0)??(c) the number must be a multiple of ?100? ?(d) the first digitsare 3742 ?? ?(e) no repetitions are? allowed?

How many? 7-digit telephone numbers are possible if the first digit cannot be two and ?(a) only even digits may be? used? ?(b) the number must be a multiple of 10? (that is, it must end in? 0)? ?(c) the number must be a multiple of ?100? ?(d) the first digits are 3742 ?? ?(e) no repetitions are? allowed?



Answers

Telephone Numbers How many 7-digit telephone numbers are possible if the first digit cannot be zero and
a. only odd digits may be used?
b. the telephone number must be a multiple of 10 (that is, it must end in zero)?
c. the telephone number must be a multiple of 100?
d. the first 3 digits are 481?
e. no repetitions are allowed?

Number 24 asks us how many four digit numbers can be formed under each condition. So a says the leading digit cannot be zero. So we have 1231234 spots. That's a four digit number. The leading digit cannot be zero. So we only have nine options because the numbers I can go here only one through nine. But in these other options, we have 10 because there we can have zero through nine, which is 10 numbers. When we multiply all these together, we get 9000. Beer says the leaving digit cannot be easier, and no repetition of digits is allowed. The 12314 options. This 1st 1 is again nine Music, not be zero in the 2nd 1 Instead of being 10 we'll also be nine because it cannot repeat the first number. So since I can't repeat the first number, you think that it would be eight. But the second digit also gets the number zero, so it's actually nine then, after that, since it can't repeat, you'll get eight and seven when you multiply it. All of those you got 4536 See says that the leading digit copies here on the number must be less than 5000. So run 23234 Since the number must be less than 5000 The only options we have for this thousands place is 12123 and four. Since it cannot be zero either. So we have four options there, and the other three numbers can be whichever numbers, um, out of 10 because they don't give any restrictions on those and d o. When you multiply these, you got 4000. And do you? They tell us that the leading digit copies here and the number of this be even. So you get it for this first pot will have nine because it can only be from one through nine. And this next spot will have 10 this next what? Love 10. Because there's no restrictions there with the last pot, since it means to you. But it can be 024246 or eight. Those Air five numbers. So this only has five options. When you multiply them all together, you got 4500

So in this problem, we're kind to create a four digit number. So in part A, all we're told is that the leading digit can not be equal to zero. So if I have my four digits and the first one cannot be zero, that means it could be 12345678 or nine. So there's nine ways we could have that first digit. Well, because our only restriction is that the first one can't be zero. That means in each of the rest of the positions, it could be the numbers zero through nine. So there's 10 ways for the second digit to get picked. 10 for the 3rd and 10 for the four. So to figure out the total combinations, we're gonna multiply nine by 10 times 10 times 10, which is equal to 9000. So there's 9000 ways to create that first number. Okay, so now let's look at Part B and says the leading digit cannot be zero and no repetition of digits is allowed. So again, we have four digits, and we're told this first digit can't be equal to zero. But again, there's nine number of ways that you can pick it, so there's nine ways for the first. Now let's think about the second. It can now be zero. The only issue is that it can't be whatever was the first number, So there's also going to be nine ways to pick the second. It could be zero, and it could be anything else except for this first digit. Okay, so now for the third number, we've already picked two numbers. So that means there's only eight left that we could choose from. And for the last digit, we've already choose. We've already chosen three numbers, so there'd only be seven choices remain. So to figure out the total ways we're gonna multiply nine times nine times eight times seven, which is 4536. So that's how many told the ways there could be to get the number in part B. Alright, Court C. So it says the leading digit cannot be zero, and the number has to be less than 5000. So because this first number has to be less than 5000, it could be 1000, 2000, 3000 or 4000. It also can't be zero, So there's four ways to pick the first number. However, there are no any of restrictions for the rest of numbers. So there's 10 ways to have the second digit 10 for the 3rd and 10 for the fourth. So there's four times 10 times 10 times 10 ways to pick this number, which is equal to 4000. Okay, so now for the last one again, we have four numbers here. So it says that the leading digit cannot be zero and the number must be even so, the only restriction on the first digit is that it can't be zero. So there's no there's nine other ways that it could be. There's no restrictions on the second or third digit, so there's 10 ways to get the 2nd 10 ways to get the third. And because this is an even number, the last number has to end with a 0246 or eight. So there's five ways we could choose that last digit. So now, to find the total, we multiply nine by 10 by 10 times five, which is equal to 4500, so there's 4500 ways to or different combinations to create that last number

Okay, So in this video, um, we are trying to answer the question, How many ray digit numbers and be informed on the condition how many three digit numbers can be found on its condition And then when we look at the conditions here, So the 1st 1 which is a so the leading the living dead, it cannot be zero. So if you write this number at the three legitimate means that you have a box like this Hey, so if you didn't know the first digit to be a the 2nd 1 be the last one, see? And you're picking from the national numbers. One, two, three war Bye. Six, 789 You know all the numbers you can think of it. The world come from this 10 numbers and we're picking on this. So for a we cannot use zero because we don't need the leading. The YouTube is zero. So that means Ah oh, yeah, it's out. So what? We have left this digits here so that I mean, for a we have nine because the remaining members tonight you have nine choices. Now we go to B B. There's a restriction being See this note restricted so we can use any of the old these numbers all 10 others. But I mean for B that are 10 ways writing be. And then there are 10 ways of writing. See? So if you remember is there's a principle called the principal. It's called the Fundamental counting Principle, which states that if two events if he won and you two are two events and you one can occur and one times or in in different ways and one different ways, you too, can occur in m em two different ways. And then the total is just the product. You just multiply them out so we can use the same thing here. So a total ah is gonna be There's got to be nine kind, miss 10. I understand. Which is, of course, 900. Now we go to me. So be, um so leading. Did it non zero. So in an zero means that the number cannot be zero. And so there are two conditions here that we need size life. So no reputations without no repetition of digit, no repetition of digits. So again we have, you know, the same box is in part a like this. So first you give us a second and let this be the last one to see now because just like before, just because the link leading did you cannot be zero. So we are nine choices because we used we could weaken, you know, pick 1 to 9. So there are nine choices here now, When we go to be, we have 10. We have nine digit left, including zero now, because we use one. Did it already? We can use that when we use that. Did it? It's out of the question. So we have nine left. So for be there are 90. It left the same to see. Ah, see, because we used to dig. It's already, uh, 10 minutes to eight. So did many. Once they're gonna be age, so they will have nine choices. Be, uh, nine choices and then see a choice. So when we calculate the total just like in part a using the using their fundamental count in principle, we have that the total is just a product. Right? So nine times nine times eight. So 99 81 when I was eight, which will give you 648 So decide that you will get 648 different numbers. Phew! For this criteria here that's given to you here. So that's part B. Now we go to see So see, uh, which is a little bit more involving. So again, the criteria perceives that. Ah, so leading digit and non zero. So that's their first, uh uh, condition. And then and so the entire number entire number. He's a multiple. Multiple of five. So again, less. Pick this box here again. The 1st 1 is a seconds be last one to see now again, uh, using previous party apart. See, we know that a automatically will have nine choices because you can use zero and then because B there's no restriction be could be anything from zero from zero tonight so that our 10 choices will be now the last one to see because we need your number to be a multiple of five. So a number. It's a multiple of another number if it is divisible by that number. So and for example, if you have you know ah, 75. So this one's a multiple of five because it's a five of the end if you have 61 This is not a multiple fire because I can't divide this next one by five. So So I mean the last digit for it to be a multiple fire. We need the last digit to end in a five or a zero. So that means off all numbers from 0 to 29 You only have two choices either at zero or a fire. So there are two choices here so that I mean the total will be nine. Time is 10 have his two. We just want not 180 1 80 numbers and then lastly for D the only restriction, buddy, is that the number number at least 400 so that no one is restricted. We have for dif now if we again right the number in a box like this like before So a B c Now, when we're picking from zero on 234 five, 67 eight Not when we pick up with this number Now dis leading to get a If that number is zero, it doesn't matter what you put for B and C. That number will always be less than 400 which we do not want you put one the same. You put two of the same you put three seeing. So all this four numbers year, they're out of the question. So we cannot use this for that four If I use three, even if I put the remaining number to be 99 there's gonna be 3 99 So that 3 99 we'll always be lesson 400 we need a minimum number to be 400. So that means that our six choices here for a because we can pick from four all the way tonight. And then, of course, would be and see there's no restriction. We can put any number. We can put 06 so that our 10 choices for P the same to see their 10 choices. So that means a total Ah, it's going to six times. 10 times 10. Did you think was this? 600 So you'll get 600 different numbers and that's the end of it.

Okay, So in this video, we're trying to hands of the question How many three digit numbers can be formed under a condition. Ah, for part A. We need the living Did it to be non zero. So again, that three digit number can be represented by X. Why said in that order now it seems we're choosing from 0 to 9. Oh, these numbers are gonna be 10 now if we exclude zero. There were nine numbers left in their place. Ah, because X is the leading digit. We do not need the rock. So that means they're gonna be nine choices for why there is not a stricken. So it's gonna be 10 for that. And other Stricken says they're gonna be tennis. Well, so total would be nine. Time is 10 times 10 with just 900. If you're wondering why I multiplied its out, I used something called the fundamental principal counting or fundamental count in principle, I would say that if n b A 22 events, if a kind of can occur M times we can occur in time, then boss even can occur in em time and time. So you just multiply. Um the numbers? No, for B B, we need the leading to get to be none. Zero and no repeat. So again, we have ex wives edge. So we're choosing from 0 to 9. The first thing you cannot be zero. So we have nine choices for X? No. For why? So this is actually X now for why? So we we're choosing from from zero tonight, excluding excluding X value they did do is selected for X. So that means I'm gonna have nine choices here now for that, um, we already used to values Rex and wives who exclude those. Then, um, anyone's gonna be 10 minutes eight miles to which is eight so multiplying again. You get nine times nine times eight, which is equal to 648. We'll see the leading did it cannot be zero, and the number has to be a multiple of five. Now, again, we have X. Why? Said so for the first digit X, we can't use zero, so we have 129 available to pick from. So that means we have nine choices now for wide in the restriction, so we can use 10 choices there now for that to ensure that this number is divisible by my five. So that has to be zero or five. So a number is a multiple of another number if that number is divisible by the number you're looking at. So, for example, 25 is even by five 26. It's not David's with my six because we need violence by five. There will be a remainder. So for that the only options we have his zero and five to each other's number is a multiple off. So that's two multiplying sound. We get nine times 10 times to giving us 1 80 now for D, so you need the number to be greater than or equals two 400? No, If this is my books of digits Ah, you can really see that if we're picking from 0123 for 56789 and we're picking from this number. If I put extra busy, you know, or one or two or three it doesn't matter what I put out for white consent. So, for example, we will put extra B three if I put why and they're to be 99 I get 3 99 This is not going to satisfy this condition here, so that means we exclude, uh, 0 to 4. Meaning I was choices for extra gonna be six. Of course. You know the Stricken. But why is that? So there's gonna be 10 charges and 10 choices. So multiply this hour, you get 600 and that's the end of it. Yeah.


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