24 questions, 19 of them short answer (type the number, as in IPMAT Indore). Each is marked with the exam level it suits. Answer, then open the step-by-step solution.
In how many distinct arrangements of the letters of SUCCESS do all the vowels come together?
Treat the vowels EU as one block: arrange 5 consonants and the block = 60 ways.
Arrange the vowels inside the block: 2 ways.
Total = 120.
A committee of 5 is chosen from 4 men and 6 women. In how many ways can it include at least one woman?
All committees: C(10, 5) = 252. All-men committees: C(4, 5) = 0.
At least one woman: 252.
Using the digits 1, 5, 7, 8 with no digit repeated, how many 3-digit numbers can be formed that are even?
The last digit must be even.
Count case by case, by last digit: ending in 8: 6.
Total = 6.
All arrangements of the letters of SMART are listed in dictionary order. What is the rank of SMART?
Alphabetical order of the letters: A, M, R, S, T.
Words that come before it: 3 × 4! = 72 (position 1: letters before S among AMRST); 1 × 3! = 6 (position 2: letters before M among AMRT).
Rank = 78 + 1 = 79.
How many solutions does x + y + z + w = 6 have in integers with each variable at least 1?
Put each variable = 1 + a new non-negative variable: the new ones add to 2.
Non-negative solutions of a sum of 4 variables equal to 2: C(2 + 3, 3) = 10.
There are 11 points in a plane, of which exactly 5 are collinear. How many straight lines can be drawn through pairs of them?
C(11, 2) = 55 if no three were collinear.
The 5 collinear points give one line instead of C(5, 2) = 10.
Lines = 55 − 10 + 1 = 46.
Using the digits 2, 7, 8, 9 with no digit repeated, how many 4-digit numbers can be formed that are odd?
The last digit must be odd.
Count case by case, by last digit: ending in 7: 6, ending in 9: 6.
Total = 12.
In how many distinct arrangements of the letters of ASSESS do all the vowels come together?
Treat the vowels AE as one block: arrange 4 consonants and the block = 5 ways.
Arrange the vowels inside the block: 2 ways.
Total = 10.
A committee of 3 is chosen from 9 men and 8 women. In how many ways can it include at least one woman?
All committees: C(17, 3) = 680. All-men committees: C(9, 3) = 84.
At least one woman: 596.
How many solutions does x + y + z = 10 have in integers with each variable at least 1?
Put each variable = 1 + a new non-negative variable: the new ones add to 7.
Non-negative solutions of a sum of 3 variables equal to 7: C(7 + 2, 2) = 36.
A committee of 5 is chosen from 5 men and 7 women. In how many ways can it have exactly 3 men?
Choose 3 of 5 men and 2 of 7 women: C(5, 3) × C(7, 2) = 10 × 21 = 210.
Moving only right or up along a grid, how many shortest paths go from (0, 0) to (7, 5) passing through (6, 2)?
(0, 0) to (6, 2): C(8, 6) = 28.
(6, 2) to (7, 5): C(4, 1) = 4.
Total = 112.
How many solutions does x + y + z + w = 13 have in integers with each variable at least 1?
Put each variable = 1 + a new non-negative variable: the new ones add to 9.
Non-negative solutions of a sum of 4 variables equal to 9: C(9 + 3, 3) = 220.
In how many ways can 8 people sit around a round table?
Fix one person to remove rotations; arrange the rest.
(8 − 1)! = 5040.
In how many ways can 8 people sit around a round table if two particular people must sit together?
Treat the pair as one unit: (7 − 1)! = 720 arrangements.
The pair can swap seats: × 2 = 1440.
All arrangements of the letters of EXAM are listed in dictionary order. What is the rank of EXAM?
Alphabetical order of the letters: A, E, M, X.
Words that come before it: 1 × 3! = 6 (position 1: letters before E among AEMX); 2 × 2! = 4 (position 2: letters before X among AMX).
Rank = 10 + 1 = 11.
Moving only right or up one unit at a time, how many shortest paths are there from (0, 0) to (6, 5)?
Each path is an arrangement of 6 R's and 5 U's.
C(11, 6) = 462.
There are 10 points in a plane, of which exactly 5 are collinear. How many triangles can be formed with these points as vertices?
C(10, 3) = 120 triples in all.
Collinear triples make no triangle: C(5, 3) = 10.
Triangles = 110.
How many distinct arrangements of the letters of INDORE begin with I?
Fix I first and arrange the remaining letters DENOR.
= 120.
Using the digits 0, 1, 6, 7 with no digit repeated, how many 3-digit numbers can be formed that are even?
The first digit cannot be 0, and the last digit must be even.
Count case by case, by last digit: ending in 0: 6, ending in 6: 4.
Total = 10.
There are 9 points in a plane, of which exactly 5 are collinear. How many straight lines can be drawn through pairs of them?
C(9, 2) = 36 if no three were collinear.
The 5 collinear points give one line instead of C(5, 2) = 10.
Lines = 36 − 10 + 1 = 27.
Moving only right or up along a grid, how many shortest paths go from (0, 0) to (7, 6) passing through (2, 5)?
(0, 0) to (2, 5): C(7, 2) = 21.
(2, 5) to (7, 6): C(6, 5) = 6.
Total = 126.
All arrangements of the letters of NOTES are listed in dictionary order. What is the rank of NOTES?
Alphabetical order of the letters: E, N, O, S, T.
Words that come before it: 1 × 4! = 24 (position 1: letters before N among ENOST); 1 × 3! = 6 (position 2: letters before O among EOST); 2 × 2! = 4 (position 3: letters before T among EST).
Rank = 34 + 1 = 35.
In how many ways can 6 people sit around a round table if two particular people must sit together?
Treat the pair as one unit: (5 − 1)! = 24 arrangements.
The pair can swap seats: × 2 = 48.
These are original practice questions written by Target Campus in the pattern, syllabus and difficulty of IPMAT Indore, IPMAT Rohtak, JIPMAT, NPAT and SET. They are not previous-year papers.
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