Permutations and Combinations — JEE Main · 2008 · 27 Apr 2008 · Offline Question with Solution

In a shop there are five types of ice-creams available. A child buys six ice-creams.

Statement-1: The number of different ways the child can buy the six ice-creams is

^{10} C_{5}

.
Statement -2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row.

(A)Statement -1 is false, Statement -2 is true✓ Correct
(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.

Answer: (A)

Solution

Buying six ice-creams from five available types is the same as choosing how many of each type to take, i.e. the number of non-negative solutions of

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6

. By stars and bars this is

(5 - 1 6 + 5 - 1) = (4 10) = 210

, not

(5 10) = 252

, so the first statement's figure is wrong and that statement is false. The second statement counts arrangements of six A's and four B's as

\frac{10 !}{6 ! 4 !} = (4 10)

, which is the correct number and so is true. The first statement is false while the second is true.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option

JEE Main · 2008 · 27 Apr 2008 · Offline · Permutations and Combinations

Solution

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