In a shop there are five types of ice-cream available. A child buys six ice-cream.
Statement - 1: The number of different ways the child can buy the six ice-cream is $${}^{10}{C_5}$$.
Statement - 2: The number of different ways the child can buy the six ice-cream is equal to the number of different ways of arranging 6 A and 4 B's in a row.
A
Statement - 1 is false, Statement - 2 is true
B
Statement - 1 is true, Statement - 2 is true, Statement - 2 is a correct explanation for Statement - 1
C
Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1
D
Statement - 1 is true, Statement - 2 is false
Explanation
Note : n items can be distribute among p persons are $${}^{n + p - 1}{C_{p - 1}}$$ ways.
Here n = 6 ice-cream
p = 5 types of ice-cream
Each ice-cream belongs to one of the 5 ice-cream type. So chosen 6 ice-crean can be divide into 5 types of ice-cream.
$$ \therefore $$ The number of different ways the child can buy the six ice-cream is = $${}^{6 + 5 - 1}{C_{5 - 1}}$$ = $${}^{10}{C_4}$$
$$ \therefore $$ Statement - 1 is false.
Number of different ways of arranging 6 A and 4 B's in a row
= $${{10!} \over {6!4!}} = {}^{10}{C_4}$$
$$ \therefore $$ Statement - 2 is true.
2
AIEEE 2007
MCQ (Single Correct Answer)
The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus $$A \cup B \cup C = S,\,A \cap B = B \cap C = A \cap C = \phi $$. The number of ways to partition S is
At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is
A
5040
B
6210
C
385
D
1110
Explanation
A voter can give vote to either 1 candidate or 2 candidates or 3 candidates or 4 candidates.
Case 1 : When he give vote to only 1 candidate then no ways = $${}^{10}{C_1}$$
Case 2 : When he give vote to 2 candidates then no ways = $${}^{10}{C_2}$$
Case 3 : When he give vote to 3 candidates then no ways = $${}^{10}{C_3}$$
Case 4 : When he give vote to 4 candidates then no ways = $${}^{10}{C_4}$$
So, total no of ways he can give votes
= $${}^{10}{C_1} + {}^{10}{C_2} + {}^{10}{C_3} + {}^{10}{C_4}$$
= 385
Note : Here we use addition rule as he can vote any one of those four rules. Whenever there is "or" choices, we use addition rule.
4
AIEEE 2005
MCQ (Single Correct Answer)
If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
A
601
B
600
C
603
D
602
Explanation
Questions Asked from Permutations and Combinations
On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions