Tips on Cracking Aptitude Questions Related To Combinations

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TIPS on cracking Aptitude Questions on
Combinations
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Important Formulae

• The number of ways of choosing r items from a
  total of n items, where the order of the chosen
  items does not matter is denoted by nCr or C(n,
  r)
• nCr or C(n, r) P(n, r) n!
• r! r! (n r)!
• C(n, 0) C(n, n) 1 We can choose all
  available items or none of the items in just one
  way
• C(n, r) C(n, n r) Choosing r items to be
  included choosing (n-r) items for exclusion
• If C(n, x) C(n, y), then either x y or x y
  n
• C(n, r) C(n, r 1) C(n 1, r) Suppose
  there are n1 items available and r items need to
  be selected. The n1 items can be grouped into
  two sets set A with n items and set B with just
  1 item. Number of ways of choosing r items from n
  1 items Number of ways of choosing r items
  when 1 item in set B is always excluded Number
  of ways of choosing r items when 1 item in set B
  is always included
• Question If C(n, r) 84, C(n, r 1) 36 and
  C(n, r1) 126, find the value of r.
• Solution
• C(n, r) ? n r 1 ? 84 ? 7
  
• C(n, r 1) r 36 3
• Again, C(n, r 1) ? n r ? 126
  ? 3
• C(n, r) r 1
  84 2

Dividing P(n, r) by r! since the order of the
chosen r items does not matter
Solving the two equations gives r 3.
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Selecting One or More Items
When each item is unique Suppose there are n
items and you can choose any number of items.
This implies that for each item, there are 2
choices choose that item or reject it. The
total number of choices 2 2 2 n times
2n Total number of ways of choosing any number
of items from a set of n items nC0 nC1
nC2 nC3 . nCn 2n Question There are
7 questions in a question paper. In how many ways
can a boy solve one or more questions? Solution
Number of ways in which at least one question
can be selected 27 1 127 Subtracting 1
because the choice of selecting zero questions is
not available When items are not unique Let
there be n things out of which p items are alike
of one kind, q items are alike of a second kind,
r items are alike of a third kind, and so on. For
all p items of the same kind, we can choose 0, 1,
2 .. or p items (Number of choices is p1). Total
number of ways of selecting any number of items
(p1)(q1)(r1). Question In how many ways can
you choose the letters of the sentence Daddy did
a deadly deed ? You can also select no letters
at all. Solution There are 9 Ds, 3 As, 3 Es,
2 Ys, 1 I and 1 L. Total number of selections
(91)(31)(31)(21)(11)(11) 1920
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Dividing items into Groups
Suppose we have p q r items that are to be
divided into 3 groups, first of size p, second of
size q and the third of size r. Number of
arrangements of all items (p q r)! However,
the order within each group does not matter. The
number of ways in which p q r items can be
divided into groups containing p, q and r items
respectively (p q r)! p! q!
r! Question In how many ways can 52 playing
cards be distributed to 4 players, giving 13
cards to each? Solution The no. of ways 52! /
(13!)4.
p q r items
?
?
?
?
?
?
?
?
?
?
?
?
Group with p items
Group with q items
Group with r items
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Simultaneous Permutations Combinations
Question How many different words, each
containing 2 vowels and 3 consonants, can be
formed using all the vowels and 17
consonants? Solution There are 5 vowels and 17
consonants in all. 2 vowels can be chosen in 5C2
ways and 3 consonants can be chosen in 17C3
ways. Thus, the letters can be selected in 5C2 x
17C3 ways. Now, each group of 5 words can be
arranged in 5! Ways. Hence, total no. of words
5C2 x 17C3 x 5! 816000. Question Out of 3
books on Economics, 4 on Political Science and 5
books on Geography, how many collections can be
made if each collection consists of at least one
book on each subject? Solution No. of ways of
choosing books on Economics 23 1 7. No. of
ways of choosing books on Political Sciences 24
1 15. No. of ways of choosing books on
Geography 25 1 31. Therefore, total number of
collections that can be formed 7 x 15 x 31 3255.
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