This post had given a series of
questions with incorrect solutions. Given below are the "debugged"
solutions to the first three questions.
1. Five
boys need to be allotted to 4 different rooms such that each boy is allotted a
room and no room is empty. In how many ways can this be done?
Given solution
Let the
boys be A, B, C, D and E. Let the rooms be 101, 102, 103 and 104. Now, we know
that exactly one room will have two occupants. First, let us try to send 4 boys
to 4 rooms, we can worry about the fifth occupant later on. Let us select 4 out
of the 5 boys first. This can be done in 5C4 ways. Now, these 4 can be allotted
to 4 different rooms in 4! ways. So, 4 boys in 4 rooms can be done in 5C4 * 4!
= 5 * 24 = 120 ways.
Now, the
fifth boy has to go into one of the rooms. He can do this in 4 ways as there
are 4 different rooms available. So, total number of outcomes = 120 * 4 = 480.
Bug in the solution:
We end up
double-counting here.
A, B, C
and D could be allotted rooms 101, 102, 103 and 104 in that order. Post this, E
could “double-up” with A. So, we would have A and E in 101, B in 102, C in 103
and D in 104.
In
another scenario, E, B, C, D could be allotted 101, 102, 103 and 104 in that
order. Post this, A could “double-up” with E. So, we would have A and E in 101,
B in 102, C in 103 and D in 104.
The two
scenarios mentioned above are identical. However, we end up counting both. This
is the reason for the double count.
Note that
in our method, we end up having a ‘first’ occupant for a room and a ‘second’
occupant. We say, A goes into room number 101 and then E ‘joins’ him. The moment
you do that, ‘order’ creeps in. We end up factoring in order when we shouldn’t.
Awesome, isn’t
it.
Correct solution:
The boys
should be allotted into rooms as 2 + 1 + 1 + 1. As in, exactly one room has 2
occupants. Some two guys have to be room-mates. Let us first select these two
room-mates. This can be done in 5C2 ways. After we have
selected these two room-mates, we practically need to just allot these 4 ‘groups’
into 4 rooms. This can be done in 4! Ways.
Total
number of options = 5C2 * 4! = 10 * 24 = 240 ways.
2. How
many 4-digit numbers with 4 distinct digits are possible?
Given solution
Let the
4-digit number be 'abcd'. Now,
'd' can
take 10 possible values - 0 to 9.
'c' can
take 9 possible values - 0 to 9 except d
'b' can
take 8 possible values - 0 to 9 except d and c
'a' can
take 6 possible values - 1 to 9 except d, b and c
So, total
number of possibilities = 6 * 8 * 9 * 10 = 4320.
Bug in the solution:
The simple
way of stating the bug is as follows “But one of b, c, d might have been zero”.
When we
say a can take 6 possible values. We eliminate 0, b, c, and d from the list 0
to 9. If b, c or d were zero, then that in that case, a will still have 7
possible options.
In this
question, the leading digit has a constraint. So, start with that. No point factoring
in the constraint in the last step. Get that out of the way first.
Correct solution:
Let the
4-digit number be 'abcd'. Now,
'a' can
take 9 possible values - 1 to 9.
'b' can
take 9 possible values - 0 to 9 except a
'c' can
take 8 possible values - 0 to 9 except a and b
'd' can
take 7 possible values - 1 to 9 except a, b and c
Total
number of options = 9 * 9 * 8 * 7.
3. In how
many ways can we rearrange the letters of the word TWOIIM such that the vowels
appear together?
Given solution
Let us
take the three vowels together and call it as X. Now, we need to rearrange the
letters of the word TWMX. This can be done in 4! ways. Or, there are 24 ways of
rearranging the letters of the word TWOIIM with vowels appearing together.
Bug in the solution:
We do not
account for the different variations that ‘X’ can come in. Now, the three
vowels have been replaced with X. Or, X is nothing but OII in some order. The
key point here is that X could be OIO, IOO or OOI. X can take three different
forms. The solution has overlooked this.
Correct solution:
Let us
take the three vowels together and call it as X. Now, we need to rearrange the
letters of the word TWMX. This can be done in 4! ways.
Now, the
three vowels have been replaced with X. Or, X is nothing but OII in some order.
The key point here is that X could be OIO, IOO or OOI. X can take three
different forms.
So total
number of possibilities = 4! * 3 = 72 ways.