JEE MAIN - Mathematics (2008 - No. 21)
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.
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.
Statement - 1 is false, Statement - 2 is true
Statement - 1 is true, Statement - 2 is true, Statement - 2 is a correct explanation for Statement - 1
Statement - 1 is true, Statement - 2 is true, Statement - 2 is not a correct explanation for Statement - 1
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.
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.
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