Answers to
Permutation, Combination, or Neither
Some Reminders Before We Start
Permutations:
n Pr
n Pr
=
n!
(n − k)!
is a formula that counts
‘The number of ways to select r objects out of n distinct objects when the order you are
selecting the objects matters, and when you can’t repeat objects’
And/Or
‘The number of ways to label r objects from a group of n distinct objects when the labels
are all different, and when you can’t repeat objects’
———————————————
Combinations:
n Cr
n Cr
=
n Cr
r!
=
n!
r! · (n − r)!
is a formula that counts
‘The number of ways to select r objects out of n distinct objects when the order you are
selecting the objects does not matter, and when you can’t repeat objects.’
And/Or
‘The number of ways to label r objects from a group of n distinct objects when all the
labels are the same, and when you can’t repeat objects.’
These formulas count specific things. You have to apply them correctly, otherwise you’re
guaranteed to get wrong answers.
1.
2.
23 P3 : We are trying to count ‘The number of ways to select 3 contestants out of 23
contestants when the order we are selecting the objects in matters, and we can’t repeat
objects’ (I can’t give Jill first place AND second place)
13 C6 :
We are trying to count ‘The number of ways to select 6 friends out of 13 friends
when the order we are selecting the friends does not matter, and we can’t repeat
friends.’ (I can’t put Tommy on my team twice)
3.
17 P8 : We are trying to count ‘The number of ways to label 8 friends from a group of
17 distinct objects when the labels are all different (Short Stop, 2n d Base, etc), and
when you can’t repeat objects’ (I can’t make one friend Catcher and Right Fielder)
4.
We are trying to count ‘The number of ways to label 20 students from a group
of 30 students when all the labels are ’Senator’, and when you can’t repeat objects.’
(I can’t label Sarah more than once)
30 C20 :
5. Neither: The Labels are President, Vice-President, Secretary, Senator, Senator, Senator, Senator, . . . . Some but not all of the labels are the same, so we can’t use either
of the simple formulas n Pr or n Cr . (Later in this chapter, you will be expected to know
how to answer this question fully)
6.
We are trying to count ‘The number of ways to label 3 students from a group
of 30 students when the labels are all different (Prez, V.P, Sec.), and when you can’t
repeat objects’ (I can’t make one student both the V.P and the secretary)
7.
48 C3 : We are trying to count ‘The number of ways to select 3 employees out of 48
employees when the order we are selecting the employees does not matter, and we can’t
repeat friends.’ (I can’t give Leana two pairs of tickets)
8.
30 P3 :
48 P3 :
We are trying to count ‘The number of ways to label 3 employees from a group
of 48 employees when the labels are all different (Country Tickets, Reggaeton Tickets,
Punk Tickets), and when you can’t repeat objects’ (I can’t give one employee tickets
to both the Reggaeton and Punk concerts)
9. Neither: 24,880 and 60,402 are both ‘5 digit numbers can be made from the the digits
0,2,4,6,8’, but both have digits that repeat, so we can’t use either formula. (You should
already know how to answer this question in a way that doesn’t use permutations or
combinations.)
10. Neither: 11,395 and 59,757 are both ‘5 digit numbers can be made from the the digits
1,3,5,7,9’, but both have digits that repeat, so we can’t use either formula. (You should
already know how to answer this question in a way that doesn’t use permutations or
combinations.)
11. Neither: 11,294 and 29,257 are both ‘5 digit numbers’, but both have digits that
repeat, so we can’t use either formula. (You should already know how to answer this
question in a way that doesn’t use permutations or combinations.)
12.
26 P5 :
We are trying to count ‘The number of ways to select 5 letters out of 26 letters when the order we are selecting the objects in matters (ALIVE is different from
AVILE), and we can’t repeat letters’
13. Neither: BOOKS and AZAYK are both “5 letter long words are there there using the
26 letters of the english alphabet”, but both have letters that repeat, so we can’t use
either formula. (You should already know how to answer this question in a way that
doesn’t use permutations or combinations.)