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1

### AIEEE 2011

These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:
A
$$N \le 100$$
B
$$100 < N \le 140$$
C
$$140 < N \le 190\,$$
D
$$N > 190$$

## Explanation

We need 3 points to create a triangle. With 10 points number of triangle possible $${}^{10}{C_3}$$

Here 6 points are on the same line so we can't make any triangle with those 6 points.

So subtract $${}^{6}{C_3}$$.

$$\therefore$$ $$N = {}^{10}{C_3} - {}^6{C_3}$$

$$= {{10\,.\,9\,.\,8} \over {1\,.\,2\,.\,3}} - {{6\,.\,5\,.\,4} \over {1\,.\,2\,.\,3}}$$

$$= 120 - 20$$

$$= 100$$

2

### AIEEE 2010

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is
A
36
B
66
C
108
D
3

## Explanation

Thus number of ways $$= ({}^3{C_2}) \times ({}^9{C_2}) = 3 \times {{9 \times 8} \over 2} = 108$$

3

### AIEEE 2009

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :
A
at least 500 but less than 750
B
at least 750 but less than 1000
C
at least 1000
D
less than 500

## Explanation

From 6 different novels 4 novels can be chosen = $${}^6{C_4}$$ ways

And from 4 different dictionaries 1 can be chosen = $${}^3{C_1}$$ ways

$$\therefore$$ From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary can be chosen = $${}^6{C_4} \times {}^3{C_1}$$ ways

Let 4 novels are N1, N2, N3, N4 and 1 dictionary is D1.

Dictionary should be in the middle. So the arrangement will be like this

_ _ D1 _ _

On those 4 blank places 4 novels N1, N2, N3, N4 can be placed. And 4 novels can be arrange $$4!$$ ways.

$$\therefore$$ Total no of ways = $${}^6{C_4} \times {}^3{C_1}$$$$\times 4!$$ = 1080
4

### AIEEE 2008

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
A
$$8.{}^6{C_4}.{}^7{C_4}$$
B
$$6.7.{}^8{C_4}$$
C
$$6.8.{}^7{C_4}$$.
D
$$7.{}^6{C_4}.{}^8{C_4}$$

## Explanation

This problem is solved using gap method. As here no 'S' is adjacent to each other so we have to put them in the gap. So first write all the letters other than 'S' such a way that there is a gap between two letters.

Given word is MISSISSIPPI.

Here, I = 4 times, S = 4 times, P = 2 times, M = 1 time

_M_I_I_I_I_P_P_

Those seven letters M, I, I, I, I, P, P can be arranged in $${{7!} \over {4!2!}}$$ ways

Those seven letters creates 8 gaps and we have to choose 4 gaps from those 8 gaps to put those four 'S' letters.

This can be done $${}^8{C_4}$$ ways.

After placing those four 'S' letters we can arrange them in $${{4!} \over {4!}}$$ ways.

Therefore, required number of words

$$= {{7!} \over {4!2!}} \times {}^8{C_4} \times {{4!} \over {4!}}$$

$$= {{7\,.\,6!} \over {4!4!}} \times {}^8{C_4}$$

$$= 7\,.\,{}^6{C_4}\,.\,{}^8{C_4}$$

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