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- Question 18 and 19, Exercise 6.2
- n. ** We must make $4$ digit numbers to keep the number less that $10000$\\ and digit at unit place must be either $3$ or $5$ to make number odd.\\ Possible numbers starting with $0$ and end... than $10000$ using all $5$ digits$=6+6=12$\\ Odd number ending at $3$ using $4$ digits out of given $5$ digits $={ }^{4} P_{3}=2$\\ $4$ digit odd number end at $5$ using $4$ digits out of$5={ }^{4} P_{3
- Question 4 and 5, Exercise 6.2
- digits out of given $6$ digits to make a F3-digit number.\\ To ensure the created number is even we have to choose the right most digit of number to be even.\\ Case $\mathrm{I}:$ If unit digit (right most digit) of number is $2$.\\ $$\underline{ },\underline{ },\underlin
- Question 1, Review Exercise 6
- llapsed="true">(a): $r!$</collapse> iv. The total number of $6$-digit number in which all the odd and only odd digits appear is:\\ * (a) $\dfrac{5}{2}\,\,6!$\... llapse> v. Let $A=\{1,2,3,4,...,20\}. $ Find the number of ways that the integer chosen a prime number is:\\ * (a) $3$ * (b) $5$ * %%(c)%% $7$
- Question 6(vi-ix), Exercise 6.1
- \in N$: $ (n!+1)$ is not divisible by any natural number between $2$ and $n$. ** Solution. ** We know $$... 2)\cdots 3.2.1$$ Hence $n!$ is divisible by every number between $1$ and $n$.\\ $n!$ can also divides by any natural number between $2$ and $n$.\\ For $(n!+1)$, $1$ is not divisible by any natural number between $2$ and $n$.\\ So $ (n!+1)$ is not divisi
- Question 7 and 8, Exercise 6.3
- i)===== There are $10$ points on circle. Find the number of lines? ** Solution. ** For a line, we need only two points so number of ways to choose $2$ points out of $10$ are $={ ... _{2}=45$ ( (ii) For triangle we need 3 points and number of ways to choose 3 points out of 10 are $={ }^{1... i)===== There are $10$ points on circle. Find the number of triangles that can be drawn? ** Solution. **
- Exercise 6.2 (Solutions)
- & 5 ]] **Question 5.** How many 7 -digits mobile number can be made using the digits 0 to 9 , if each number starts with 5 and no digit is repeated?\\ [[math-11-... on: Question 10 & 11]] **Question 12.** Find the number of arrangement of letters of the word VOWEL in wh... on: Question 18 & 19]] **Question 20.** Find the number of ways that 6 men and 6 women seated at a round
- Exercise 6.3 (Solutions)
- on 8.** There are 10 points on a circle. Find the number of $(\mathrm{i})$ lines (ii) triangles that can b... ution: Question 7 & 8]] **Question 9.** Find the number of diagonals in n sided polygon?\\ [[math-11-nbf... -p8|Solution: Question 9 & 10]] **Question 11.** Number of diagonals in $n$-sided polygon is 35 . Find the number $n$ ?\\ [[math-11-nbf:sol:unit06:ex6-3-p9|Solutio
- Question 4, 5 and 6, Review Exercise 6
- utations starting with $0$ results into $5$ digit number,\\ and number of such permutations is $$5!=120$$ Number of $6-$digits numbers formed $$=720-120=600$$ =====Question 5=... eve persons are seated at a round table. Find the number of ways of their arrangement if two particular pe
- Question 20 and 21, Exercise 6.2
- mabad, Pakistan. =====Question 20===== Find the number of ways that $6$ men and $6$ women seated at aaro... ng words:\\ WHY, SAD, TWO, MADE ** Solution. ** Number of words $= n = 4$ Number ot permuations of 4 things taking 4 at a time $= {}^4P_4 = 24.$ GOOD ====Go
- Question 9 and 10, Exercise 6.3
- amabad, Pakistan. =====Question 9===== Find the number of diagonals in $n$ sided polygon? ** Solution. ... are sides of polygon and rest are diagonals.\\ So number of diagonals of $n$-sided polygon $={ }^{n} C_{2}... $ girls can be chosen in only one way. So\\ Total number of ways to divide girls into two groups of 3 and
- Question 16 and 17, Exercise 6.2
- ** Solution. ** If digit at units place is odd, number is odd.\\ We shall fix unit place with $1, 3$ or ... and calculate arrangements of remaining digits.\\ Number of six digit odd numbers $=3 \times{ }^{5} P_{5}=
- Question 11 and 12, Exercise 6.3
- ard, Islamabad, Pakistan. =====Question 11===== Number of diagonals in $n$-sided polygon is $35$. Find the number $n$? ** Solution. ** So given that \begin{alig
- Exercise 6.1 (Solutions)
- \ (viii) $(n!+1)$ is not divisible by any natural number between 2 and $n$. (ix) $\quad(n!)^{2} \leq n^{n}
- Question 6 and 7, Exercise 6.2
- nd no digit is repeated? ** Solution. ** Total number of digits given $=7$.\\ Even places are $2^{\text
- Question 5 and 6, Exercise 6.3
- only choose $10$ players out of $15$ so-this time number of ways are $$ { }^{15} C_{10}=3003 $$ =====Que