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Question 11 Mark

Write the maximum number of points of intersection of 8 straight lines in a plane.

Answer

There are 8 straight lines,
Two intersecting lines give one point of intersection. So,
Required number of point $={^\text{8}}\text{C}_{\text{2}}$
$=\frac{8\times7}{2}$
Required number of points = 28.

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Question 21 Mark

Write the number of diagonals of an n-sided polygon.

Answer

Polygon has n sides and diagonals are kinds made by joining two opposite points = Number of total lines in polygon - Number of sides in the Polygon.
Number of diagonal $={^\text{n}}\text{C}_{\text{2}}-\text{n}$
$=\frac{\text{n}(\text{n}-1)}{2}-\text{n}$
$=\text{n}\Big[\frac{\text{n}-1}{2}-1\big]$
$=\text{n}\Big[\frac{\text{n}-1-2}{2}\Big]$
$=\frac{\text{n}(\text{n}-3)}{2}$

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Question 31 Mark

Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.

Answer

Out of 12 boys, we have to divide them into three groups of 4 boys each.
Number of ways of arrangement of 4 boys in a group = 4!
Number of ways of arrangement 5 of 12 boys in a group = 12!
Therefore,
Required number of ways $=\frac{12!}{(4!)^{3}3!}$

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Question 41 Mark

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Answer

There are two sets parallel lines, one with 4 parallel lines and with 3 parallel line. To form a parallel we should have 2 pair of parallel lines.
Required number of parallelograms $={^\text{4}}\text{C}_{\text{2}}\times{^\text{3}}\text{C}_{\text{2}}$
$=6\times3$
$=18$
Required number of parallelograms = 18.

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Question 51 Mark

Write $\sum\limits_\text{r=0}^\text{m}\ {^\text{n+r}}\text{C}_{\text{r}}$ in the simplified form.

Answer

$\sum\limits_\text{r=0}^\text{m}\ {^\text{n+r}}\text{C}_{\text{r}}$
$={^\text{n}}\text{C}_{\text{0}}+{^\text{n+1}}\text{C}_{\text{1}}+{^\text{n+2}}\text{C}_{\text{2}}+{^\text{n+3}}\text{C}_{\text{3}}+ ....+{^\text{n+m}}\text{C}_{\text{m}}$
$=\frac{\text{n}!}{0!\text{n}!}+\frac{(\text{n+1})!}{1!\text{n}!}+\frac{(\text{n}+2)!}{2!\text{n}!}+\frac{(\text{n}+3)!}{3!\text{n}!}+....+\frac{(\text{n+m})!}{\text{n}!\text{m}!}$
$=\frac{(\text{m}!)(\text{n}!)+(\text{n}+1)!\text{m}!+(\text{n+2})!\text{m}!+...+(\text{n}+\text{m})!\text{m}!}{\text{n}!\text{m}!}$
$=\frac{(\text{n}+\text{m}+1)}{\text{m}!(\text{n}+1)!}$
$={^\text{n+m+1}}\text{C}_{\text{n+1}}$

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Question 61 Mark

Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.

Answer

We have select 2 vowels out of 4 and 3 consonants out of 5 consonants and them, arrange there 5 letter.
So,
Number of ways to select 2 vowels out of 4 and 3 consonants out 5 consonants to arrange them $={^\text{4}}\text{C}_{\text{2}}\times{^\text{5}}\text{C}_{\text{3}}\times5!$

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Question 71 Mark

Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.

Answer

There are 10 red and 8 white balls.
Number of ways ti select 5 red and 4 white balls $={^\text{10}}\text{C}_{\text{5}}\times{^\text{8}}\text{C}_{\text{4}}$

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Question 81 Mark

There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.

Answer

There are 3 letters to be placed in 3 directed envelopes.
Total number of ways in which 3 letters can be placed in 3 envelops
= 3 × 2 × 1
= 6
Number of ways in which only one letter can be placed in correct envelpe = 3
Number of ways in which only 2 letter can be placed in correct envelops = Number of ways in which all there is coeerct envelops = 1
So,
Total number of ways in which no letter no letter in is correct envelops = 1
Required number of ways = 2.

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Question 91 Mark

Write the expression ${^\text{n}}\text{C}_{\text{r}+1}+{^\text{n}}\text{C}_{\text{r-1}}+2\times{^\text{n}}\text{C}_{\text{r}-1}$in the simplest form.

Answer

${^\text{n}}\text{C}_{\text{r}+1}+{^\text{n}}\text{C}_{\text{r-1}}+2\times{^\text{n}}\text{C}_{\text{r}-1}$
$=({^\text{n}}\text{C}_{\text{r}+1}+{^\text{n}}\text{C}_{\text{r}})+{^\text{n}}\text{C}_{\text{r}}+{^\text{n}}\text{C}_{\text{r-1}}$
$={^\text{n+1}}\text{C}_{\text{r}+1}+{^\text{n+1}}\text{C}_{\text{r}}$
$={^\text{n+1+1}}\text{C}_{\text{r}+1}$
$={^\text{n+2}}\text{C}_{\text{r}+1}$

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Question 101 Mark

If ${^\text{35}}\text{C}_{\text{n}+7}={^\text{35}}\text{C}_{\text{4n-2}},$ then write the values of n.

Answer

${^\text{35}}\text{C}_{\text{n}+7}={^\text{35}}\text{C}_{\text{4n-2}}$
⇒ n + 7 = 4n - 2
⇒ 3n = 9
⇒ n = 3
${^\text{35}}\text{C}_{35-(\text{n}+7})={^\text{35}}\text{C}_{\text{4n-2}}$
⇒ 35 - (n + 7) = 4x - 2
⇒ 35 - x - 7 = 4x - 2
⇒ 5x = 30
⇒ n = 6

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Question 111 Mark

Write the value of $\sum\limits_\text{r-1}^6 {^\text{56-r}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}.$

Answer

$\sum\limits_\text{r-1}^6 {^\text{56-r}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{56-1}}\text{C}_{\text{3}}+{^\text{56-2}}\text{C}_{\text{3}}+{^\text{56-3}}\text{C}_{\text{3}}+{^\text{56-4}}\text{C}_{\text{4}}+{^\text{56-5}}\text{C}_{\text{5}}+{^\text{56-6}}\text{C}_{\text{6}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{3}}+({^\text{50}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+({^\text{51}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+({^\text{53}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+({^\text{54}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{4}})$
$=({^\text{55}}\text{C}_{\text{3}}+{^\text{55}}\text{C}_{\text{4}})$
$={^\text{56}}\text{C}_{\text{4}}$

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