Permutations and Combinations Test 32

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Permutations and Combinations Test 32

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Weekly Quiz Competition

Question 1
1 / -0

Directions for questions 1 to 3: Find the related word/ letters/numbers from given alternatives.
12:72::8:?

A
36

B
32

C
38

D
40

Solution

12 $$\times \, \frac{12}{2}$$ = 72 similarly, 8 $$\times \, \frac{8}{2}$$ = 32
Question 2
1 / -0

If $$12\times 16 = 188$$ and $$14\times 18 = 248$$, then find the value of $$16\times 20 = ?$$

A
$$320$$

B
$$360$$

C
$$316$$

D
$$318$$

Solution

$$12\times 16 = 192 - 4 = 188$$
$$14\times 18 = 252 - 4 = 248$$
$$16\times 20 = 320 - 4 = 316$$.
Question 3
1 / -0

Select the missing number from the given alternatives.

A
110

B
115

C
121

D
54

Solution

48 $$\div$$ 2 = 24;
24 $$\times$$ 3 = 72: 72 $$\div$$ 2 = 36;
36 $$\div$$ 3 = 108; 108 $$\div$$ 2 = 54.
Question 4
1 / -0

Choose the correct answer from the alternatives given.
If $$1^2+2^2$$ +.... + $$x^2$$ = $$\frac{x(x + 1)(2x + 1)}{6}$$then $$1^2$$ + $$3^2$$+ $$5^2$$ + .... + $$19^2$$ is equal to

A
$$1330$$

B
$$2100$$

C
$$2485$$

D
$$2470$$

Solution

$$(1^2+3^2+5^2+...........+19^2)$$
= $$\displaystyle\, $$(1^2$$ \, + \, $$2^2$$ \, + \, $$3^2$$ \, + \, ........... \, + \, 20^2) \, - \, (2^2 \, + \, 4^2 \, + \, 6^2 \, + \, ........... \, + \, 20^2)$$
= $$\displaystyle\, (1^2 \, + \, 2^2 \, + \, 3^2 \, + \, .......... \, + \, 20^2) - 4(1 \, + \, 2^2 \, + \, 3^2 \, + \, .......... \, + \, 10^2)$$
Using the formula,
$$\displaystyle\, = \frac{20 \times 21 \times 41}{6} - 4 \left ( \frac{10 \times 11 \times 21}{6} \right ) = 2870 - 1540 = 1330$$
Question 5
1 / -0

How many different $$4$$-person committees can be chosen form the $$100$$ members of the Senate ?

A
$$25$$

B
$$400$$

C
$$3,921,225$$

D
$$94,109,400$$

Solution

Total number of members $$=100$$.
Members has to be selected $$=4$$.
$$\therefore$$ different committees $$=^{100}C_{4}=\large{\frac{100!}{4!\times96!}}$$ $$=3,921,225$$.
Question 6
1 / -0

Choose the correct answer alternatives given.
Select the missing number from the given alternatives.

A
18

B
21

C
5

D
17

Solution

80 = 8 $$\times$$ (8 + 2)
143 = 11 $$\times$$ (11 + 2)
323 = ? $$\times$$ (? + 2)
$$\therefore$$ ? = 17
Question 7
1 / -0

Arrange these numbers in ascending order.
$$756, 567, 657, 676$$

A
$$657, 567, 756, 676$$

B
$$567, 657, 676, 756$$

C
$$756, 676, 657, 567$$

D
$$676, 756, 567, 657$$

Solution

$$\Rightarrow$$ Numbers are said to be in ascending order when they are arranged from the smallest to the largest number.
$$\Rightarrow$$ The numbers which we have to arrange in ascending order are $$756,\,567,\,657$$ and $$676$$
$$\Rightarrow$$ $$567<657<676<756$$
$$\therefore$$ Ascending order $$=567,\,657,\,676,\,756$$
Question 8
1 / -0

The number of ways in which we can select 5 letters of the word INTERNATIONAL is equal to

A
$$200$$

B
$$220$$

C
$$242$$

D
$$256$$

Solution

We have thirteen letters.
$$2l, 3N, 2T, 2A$$ and $$(E, O, L,R)8$$ (types)
We can select 5(five) letters in the following manner:

1. All different $${^8C}_5 = \dfrac{8.7.5}{1.2.3} = 56$$ ways

2. 2 alike, 3 different $${^4C}_1 . {^7C}_3 = 435 = 140$$ ways (we have 4 sets of alike letters)

3. 3 alike, 2 different $${^1C}_1 . {^7C}_2 = 1.21 = 21$$ ways (we have only one set of 3 alike)

4. 3 alike and 2 alike $${^1C}_1 . {^3C}_1 = 1.3 = 3$$ ways

5.Two sets of alike and one different
$${^4C}_2 . {^6C}_1 = 6.6 = 36$$ ways

$$\therefore$$ Total number of selection is
$$56 + 140 + 21 + 3 + 36 = 256$$ ways $$\Longrightarrow$$ (d)
Question 9
1 / -0

The area of the triangle with vertices at $$(-4, 1), (1, 2)(4, -3)$$ is

A
$$14$$

B
$$16$$

C
$$15$$

D
None of these

Solution
Question 10
1 / -0

The area of the triangle whose vertices are (3,8), (-4,2) and (5,-1) is

A
$$75 \ sq. units$$

B
$$37.5 \ sq. units$$

C
$$45 \ sq. units$$

D
$$22.5 \ sq. units$$

Solution

Let $$A(3,8), B(-4,2),C(5,-1)$$ be the vertices of the given $$\triangle ABC$$.

Then,

$$(x_{1}=3,y_{1}=8),(x_{2}=-4,y_{2}=2),(x_{3}=5,y_{3}=-1)$$

Area of $$\triangle ABC$$ = $$\dfrac{1}{2}|[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]|$$

$$=\dfrac{1}{2}|3[2-(-1)]-4(-1-8)+5(8-2)|$$

$$=\dfrac{1}{2}|9+36+30|=\dfrac{75}{2}=37.5 \ sq. units$$

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Re-Attempt Weekly Quiz Competition

Permutations and Combinations Test 32

Result

Permutations and Combinations Test 32

Score

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Rank

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SHARING IS CARING

Question 1

36

32

38

40

Solution

Question 2

$$320$$

$$360$$

$$316$$

$$318$$

Solution

Question 3

110

115

121

54

Solution

Question 4

$$1330$$

$$2100$$

$$2485$$

$$2470$$

Solution

Question 5

$$25$$

$$400$$

$$3,921,225$$

$$94,109,400$$

Solution

Question 6

18

21

5

17

Solution

Question 7

$$657, 567, 756, 676$$

$$567, 657, 676, 756$$

$$756, 676, 657, 567$$

$$676, 756, 567, 657$$

Solution

Question 8

$$200$$

$$220$$

$$242$$

$$256$$

Solution

Question 9

$$14$$

$$16$$

$$15$$

None of these

Solution

Question 10

$$75 \ sq. units$$

$$37.5 \ sq. units$$

$$45 \ sq. units$$

$$22.5 \ sq. units$$

Solution

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