What is the smallest number greater than 5000
Hint: Here two concepts are used-
Probability of an event happening $ = $ Number of favorable cases/Number of total cases Combination is a selection of items from a collection, such that the order of selection does not matter. Formula- ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$, where $n$ is the number of things to choose from and we choose $r$ from them. In our question we will see two cases to form a number from a $n$ number of digits to fill $r$ number of places, when digits are repeated\[ = {}^n{C_1} \times {}^n{C_1} \times {}^n{C_1} \times ..... \times rth\]term$ = {({}^n{C_1})^r}$ when digits are not repeated\[ = {}^n{C_1} \times {}^{n - 1}{C_1} \times {}^{n - 2}{C_1} \times .....{}^{n - (r - 1)}{C_1}\] Complete answer: Show Now we have $5$digits to choose from to fill $3$ places so we have to choose $3$ times. Therefore the number of possible ways $ = {5^3} = 125$ >Then we will take $7$ as thousand’s place digit- And again we have $5$digits to choose from to fill $3$ places so we have to choose $3$ times. Therefore the number of possible ways $ = {5^3} = 125$ Finally the number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$when the digits are repeated are $ = 125 + 125 = 250$ Statement (1) (ii)To find the probability of the first case when digits are repeated, this would be the total number of cases. Now we have $5$digits to choose from to fill $2$ places so we have to choose $2$ times. Therefore the number of possible ways $ = {5^2} = 25$ >When a thousand's place digit is $5$and the one’s place digit is $0$. Now we have $5$digits to choose from to fill $2$ places so we have to choose $2$ times. Therefore the number of possible ways $ = {5^2} = 25$ >When a thousand's place digit is $7$and the one’s place digit is $5$. Now we have $5$digits to choose from to fill $2$ places so we have to choose $2$ times. Therefore the number of possible ways $ = {5^2} = 25$ >When a thousand's place digit is $7$and the one’s place digit is $0$. Now we have $5$digits to choose from to fill $2$ places so we have to choose $2$ times. Therefore the number of possible ways $ = {5^2} = 25$ Finally the number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are repeated are $ = 25 + 25 + 25 + 25 = 100$ Statement (2) To find the probability of the first case when digits are repeated, this would be the number of favorable cases. Probability of forming a number which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are repeated $ = $ Number of favorable cases/total number of cases $ = \dfrac{{100}}{{250}}$ [From statement (1) and statement (2)] $ = \dfrac{2}{5}$ (iii)The number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$when the digits are not repeated are- Now we have $4$digits for our first choice, $3$ digits for the second and $2$ for the third. Therefore the number of possible ways here$ = 4 \times 3 \times 2 = 24$ >Then we will take $7$ as thousand’s place digit- Now we have $4$digits for our first choice, $3$ digits for the second and $2$ for the third. Therefore the number of possible ways here$ = 4 \times 3 \times 2 = 24$ Finally the number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$when the digits are not repeated are $ = 24 + 24 = 48$ Statement (3) To find the probability of a second case when digits are not repeated, this would be the total number of cases. And we have to find the number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$when the digits are not repeated are also divisible by $5$. This case cannot be possible since $5$is repeated. When a thousand's place digit is $5$and the one’s place digit is$0$. Now we have $3$digits for our first choice and $2$ digits for the second. Therefore the number of possible ways here$ = 3 \times 2 = 6$ When a thousand's place digit is $7$and the one’s place digit is $5$. Now we have $3$digits for our first choice and $2$ digits for the second. Therefore the number of possible ways here$ = 3 \times 2 = 6$ When a thousand's place digit is $7$and the one’s place digit is $0$. Now we have $3$digits for our first choice and $2$ digits for the second. Therefore the number of possible ways here$ = 3 \times 2 = 6$ Finally the number of numbers which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are not repeated are $ = 6 + 6 + 6 = 18$ Statement (4) To find the probability of a second case when digits are repeated, this would be the number of favorable cases. Probability of forming a number which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are not repeated $ = $ Number of favorable cases/total number of cases $ = \dfrac{{18}}{{48}}$ [from statement (3) and statement (4)] $ = \dfrac{3}{8}$ Hence, Probability of forming a number which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are repeated $ = \dfrac{2}{5}$ Probability of forming a number which are not less than $5000$and formed by $0,1,3,5$ and $7$ also divisible by $5$when the digits are not repeated $ = \dfrac{3}{8}$ Note:The number of ways to arrange $n$ objects in a row, of which exactly $r$ objects are identical, is$\dfrac{{n!}}{{r!}}$. How many numbers greater than 5000 can be formed?So, total number of nos. Possible =4×5×4×3= 240. Q.
What is the biggest 5 digit number?Five Digit Numbers
The smallest five-digit number is 10000 and the greatest five-digit number is 99999.
What is the largest 5 digit number without?Greatest 5 Digit Number
The greatest 5-digit number is 99,999 because the number after 99,999 is 1,00,000 which becomes a 6-digit number.
Which is the smallest 4 digit number greater than 3000 all with different digits?The smallest four-digit number, using all different digits, is 1023.
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