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1

### AIEEE 2004

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is
A
$${}^8{C_3}$$
B
21
C
$${3^8}$$
D
5

## Explanation

To distribute n objects among p people where everyone should get atleast one object, then number of ways to distribute those n objects

= $${}^{n - 1}{C_{p - 1}}$$

For this question, n = 8 and p = 3

$$\therefore$$ Number of ways = $${}^{8 - 1}{C_{3 - 1}}$$ = $${}^7{C_2}$$ = 21
2

### AIEEE 2004

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order
A
480
B
240
C
360
D
120

## Explanation

In the word ''GARDEN'', there are two vowels A and E present, and A should come always before E.

$$\therefore\,\,\,$$ Total no of ways = $${{6!} \over {2!}}$$ = 360

Here A and E has fixed order that is why we divide by 2!.
3

### AIEEE 2003

If $${}^n{C_r}$$ denotes the number of combination of n things taken r at a time, then the expression $$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$$ equals
A
$$\,{}^{n + 1}{C_{r + 1}}$$
B
$${}^{n + 2}{C_r}$$
C
$${}^{n + 2}{C_{r + 1}}$$
D
$$\,{}^{n + 1}{C_r}$$

## Explanation

Arrange it this way,

$$^n{C_{r + 1}} + 2.{}^n{C_r} + {}^n{C_{r - 1}}$$

$$= {}^n{C_{r + 1}} + {}^n{C_r} + {}^n{C_r} + {}^n{C_{r - 1}}$$

$$\left[ \, \right.$$ Now use the rule,

$$\left. {{}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}} \right]$$

$$= {}^{n + 1}{C_{r + 1}} + {}^{n + 1}Cr$$

$$= {}^{n + 2}{C_{r + 1}}$$
4

### AIEEE 2003

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
A
346
B
140
C
196
D
280

## Explanation

Case 1 :

No of ways student can answer 10 questions = $${}^5{C_4} \times {}^8{C_6}$$ = 140

Case 2 :

No of ways student can answer 10 questions = $${}^5{C_5} \times {}^8{C_5}$$ = 56

$$\therefore$$ Total ways = 140 + 56 = 196

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