( Answers at the end of all questions )

Page 1

(1)

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word ‘SACHIN’ appears at serial number ( a ) 601 ( b ) 600 ( c ) 603 ( d ) 602 [ AIEEE 2005 ]

(2)

The value of

50

C4 +

55

r =1

∑

6

56

-r C 3

is

( a ) 55 C 4

(b)

C3

( c ) 56 C 3

(d)

56

C4

[ AIEEE 2005 ]

(3)

How many ways are here to arrange the letters in the word GARDEN with the vowels in alphabetical order ? ( a ) 120 ( b ) 240 ( c ) 360 (d ) 480 [ AIEEE 2004 ]

(4)

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is (a) 5 ( b ) 21 (c) 3 8

( d ) 8 C3

[ AIEEE 2004 ]

(5)

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 ) 140 ( b ) 196 ( c ) 280 ( d ) 346 [ AIEEE 2003 ]

(6)

The number of ways in which 6 men and 5 women can dine at a round table, if no two women are to sit together, is given by ( a ) 30 (b) 5! × 5! (c) 5! × 4! (d) 7! ×5! [ AIEEE 2003 ]

(7)

If n C r

denotes the number of combinations of n things taken r at a time, then the is ( d ) n + 1Cr + 1 [ AIEEE 2003 ]

value of expression n C r + 1 + n Cr - 1 + 2 n C r ( a ) n + 2 Cr ( b ) n + 2 Cr + 1 ( c ) n + 1C r

05 - PERMUTATIONS AND COMBINATIONS

( Answers at the end of all questions ) (8)

Page 2

If repetition of the digits is allowed, then the number of even natural numbers having three digits is ( a ) 250 ( b ) 350 ( c ) 450 ( d ) 550 [ AIEEE 2002 ]

(9)

If n + 1 C 3 = 2 n C 2 , then the value of n is (a) 3 (b) 4 (c) 5 (d) 6 [ AIEEE 2002 ]

( 10 ) If n Cr - 1 = 36, n Cr = 84 and n C r + 1 = 126, then n and r are respectively ( a ) 9, 6 ( b ) 9, 3 ( c ) 6, 3 ( d ) 6, 2 [ AIEEE 2002 ]

( 11 ) If ( 1 + x )

= C 0 + C1 x + C 2 x 2 + ... + Cn x n , then the value of 3C 3 C1 2C 2 nCn + + + ... + is C0 C1 C2 Cn - 1 n 2 (b) n(n + 1) (c) n ( n + 1) 12 (d) n ( n + 1) 2 [ AIEEE 2002 ]

n

(a)

( 12 ) A rectangle is constructed of lengths ( 2m - 1 ) and ( 2n - 1 ) units where m, n ∈ I and small rectangles are inscribed in it by drawing parallel lines. Find the maximum number of rectangles that can be inscribed in it having odd unit length. (a) m (c) 4 2

- n2

( b ) mn ( m + 1 ) ( n + 1 ) (d) m n

2 2

m + n - 2

[ IIT 2005 ]

2 ( 13 ) If n - 1C r = ( k - 3 ) n C r + 1 , then k lies between

(a) (-

∞, -2)

( b ) ( 2,

∞)

(c) [-

3,

3]

(d) ]

3, 2]

[ IIT 2004 ]

( 14 ) The number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently is ( a ) 40 ( b ) 60 ( c ) 80 ( d ) 100 [ IIT 2002 ]

05 - PERMUTATIONS AND COMBINATIONS

( Answers at the end of all questions )

Page 3

( 15 )

Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon on n sides. If T n + 1 - Tn = 21, then n equals (a) 5 (b) 7 (c) 6 (d) 4 [ IIT 2001 ]

n n n ( 16 ) For 2 ≤ r ≤ n, + 2 r -1 + r -2 r n + 1 (a) r -1 n+1 (b) 2 r +1

= n + 2 (d) r

n + 2 (c) 2 r

[ IIT 2000 ]

( 17 ) How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions. ( a ) 16 ( b ) 36 ( c ) 60 ( d ) 180

( 18 ) If a n =

r

∑

n

1

nC r

, then

r

=0

∑

n

r

nC r

equals

=0

(a) (n - 1)an

(b) nan

(c)

1 nan 2

( d ) none of these

[ IIT 1998 ]

( 19 )

An n - digit number is a positive number with exactly n digits. Nine hundred distinct n - digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which...