Its election time at a high school that has a total of 50
students in the junior class. How many ways can a class president,
class vice president, class treasurer,and class secretary be chosen
if each student may only hold one office?
Its election time at a high school that has a total of 50
students in the junior class. How many ways can a class president,
class vice president, class treasurer,and class secretary be chosen
if each student may only hold one office?
SOLUTION: Here the order that we choose the students is
important. The same group of four people could be given different
offices and it would count as a different result. Since order
matters, we are really trying to determine the number of
permutations of 50 taken four at a time. The formula for
permutations gives us P( 50, 4 ) = 50!/46! = 50 x 49 x 48 x 47 =
5527200.
The same class of 50 students wants to form a prom committee.
How many ways can a four person prom committee be selected from the
junior class?
SOLUTION: This is similar to the last example in that the
numbers are the same, we are choosing four people out of 50.
However this time the order is unimportant. If a group of people is
on the committee, then it does not matter what order they were
selected. Since order does not matter, we are trying to determine
the number of combinations of 50 taken four at a time. The formula
for combinations gives us C( 50, 4 ) = 50!/(4!46!) = (50 x 49 x 48
x 47)/(4 x 3 x 2 x 1) = 230300.
If we want to form a group of five students and we have 20 to
choose from, how many ways is this possible?
SOLUTION: In forming a group we dont need to worry about the
order. So this is a combination and we need to calculate C( 20, 5 )
= 20!(5!15!) = (20 x 19 x 18 x 17 x 16)/ (5 x 4 x 3 x 2 x 1) =
15504.
How many ways can we arrange four letters from the word computer
if repetitions are not allowed, and different orders of the same
letters count as different arrangements?
SOLUTION: Since were keeping track of the order of the letters,
this is a permutation. We are selecting four letters from a total
of eight, and must calculate P( 8, 4 ) = 8!/4! = 8 x 7 x 6 x 5 =
1680
How many ways can we arrange four letters from the word computer
if repetitions are not allowed, and different orders of the same
letters count as the same arrangement?
SOLUTION: Order is not important, so this is a combination. We
are selecting four letters from a total of eight, and calculate C(
8, 4 ) = 8!/(4!4!) = (8 x 7 x 6 x 5)/(4 x 3 x 2 x 1) = 70
How many different four digit numbers are possible if we can
choose any digits from 0 to 9 and all of the digits must be
different?
SOLUTION: Here the order is important, so this is a
permutations. We must calculate P( 10 , 4) = 10!/6! = 10 x 9 x 8 x
7 = 5040.
If we are given a box containing seven books, how many ways can
we arrange three of them on a shelf?
SOLUTION: Here we care about how the books are arranged, and so
the order is important. This means that we are dealing with a
permutation. We must calculate P( 7 , 3) = 7!/4! = 7 x 6 x 5 =
210.
If we are given a box containing seven books, how many ways can
we choose collections of three of them from the box?
SOLUTION: Here we care about choosing groups of three books, and
so the order is not relevant to the problem. This means that we are
dealing with a combination. We must calculate C( 7 , 3) = 7!/(4!3!)
= 7 x 6 x 5 = 35.
Question 1: In how many ways can the letters of the word ABACUS
be rearranged such that the vowels always appear together?
PRIVATE "TYPE=PICT;ALT=(4! * 3!) / 2!"
INCLUDEPICTURE \d
"images/permutation_combination_0812053.gif".Answer &
Explanation
Question 2: How many different four letter words can be formed
(the words need not be meaningful) using the letters of the word
MEDITERRANEAN such that the first letter is E and the last letter
is R? Answer & Explanation
Question 3:What is the probability that the position in which
the consonants appear remain unchanged when the letters of the word
"Math" are re-arranged? Answer & Explanation
Question 4: There are 6 boxes numbered 1, 2, ... 6. Each box is
to be filled up either with a red or a green ball in such a way
that at least 1 box contains a green ball and the boxes containing
green balls are consecutively numbered. The total number of ways in
which this can be done is: Answer & Explanation
Question 5: A man can hit a target once in 4 shots. If he fires
4 shots in succession, what is the probability that he will hit his
target? Answer & Explanation
Question 6: In how many ways can 5 letters be posted in 3 post
boxes, if any number of letters can be posted in all of the three
post boxes? Answer & Explanation
Question 7: Ten coins are tossed simultaneously. In how many of
the outcomes will the third coin turn up a head? Answer &
Explanation
Question 8: In how many ways can the letters of the word
"PROBLEM" be rearranged to make seven letter words such that none
of the letters repeat? Answer & Explanation
