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1

### WB JEE 2010

MCQ (Single Correct Answer)

The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is

A
96
B
144
C
512
D
576

## Explanation

There are 4 odd places

$$\therefore$$ 3 vowels (O, I, E) can be arranged in 4 places in $${}^4{P_3} = 24$$ ways

Remaining 4 letters (C, M, B, N) can be arranged in remaining 4 places in 4! = 24 ways

$$\therefore$$ Required number of permutations = 24 $$\times$$ 24 = 576

2

### WB JEE 2009

MCQ (Single Correct Answer)

A polygon has 44 diagonals. The number of the sides is

A
10
B
11
C
12
D
13

## Explanation

Let number of vertices of polygon = n

Total number of line segments joining two vertices = $${}^n{C_2}$$

$$\therefore$$ Number of diagonals

= Total number of line segment $$-$$ number of sides

= $${}^n{C_2}$$ $$-$$ n = 44

$$\Rightarrow {{n(n - 1)} \over 2} - n = 44 \Rightarrow {n^2} - 3n - 88 = 0$$

$$\Rightarrow {n^2} - 11n + 3n - 88 = 0 \Rightarrow n(n - 11) + 3(n - 11) = 0$$

$$\Rightarrow (n - 11)(n + 3) = 0$$

$$\Rightarrow n - 11 = 0 \Rightarrow n = 11$$ ($$\because$$ $$n \ne - 3$$)

3

### WB JEE 2008

MCQ (Single Correct Answer)

The number of ways four boys can be seated around a round table in four chairs of different colours is

A
24
B
12
C
23
D
64

## Explanation

Number of ways in which four boys can be seated around a round table = 4! = 24.

4

### WB JEE 2008

MCQ (Single Correct Answer)

How many odd numbers of six significant digits can be formed with the digits 0, 1, 2, 5, 6, 7 when no digit is repeated?

A
120
B
96
C
360
D
288

## Explanation

For odd number, 1, 5 or 7 should be on the unit place. At the lakh's place 0 can't be there, so the lakh's place can be filled by any one of four numbers.

Rest of the four middle placed can be arranged in $${}^4{P_4}$$ ways.

So, number of odd number = 4 $$\times$$ $${}^4{P_4}$$ $$\times$$ 3

= 4 $$\times$$ 4 $$\times$$ 3 $$\times$$ 2 $$\times$$ 3 = 288.

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