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  4. Box 1 contains three cards bearing numbers 1,2,3 ; box 2 contains five cards bearing numbers 1,2,3,4,5 ; and box 3 contains seven cards bearing numbers 1,2,3,4,5,6,7 . A card is drawn from each of the boxes. Let xi be the number on the card drawn from the i textth box, i=1,2,3 The probability that x1+x2+x3 is odd, is

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Q. Box 1 contains three cards bearing numbers $1,2,3 ;$ box $2$ contains five cards bearing numbers $1,2,3,4,5 ;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7 .$ A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the $i^{\text{th}}$ box, $i=1,2,3$
The probability that $x_{1}+x_{2}+x_{3}$ is odd, is

JEE AdvancedJEE Advanced 2014

Solution:

Case I : One odd, $2$ even
Total number of ways $=2 \times 2 \times 3+1 \times 3 \times 3+1 \times 2 \times 4=29$.
Case II: All $3$ odd
Number of ways $=2 \times 3 \times 4=24$
Favourable ways $=53$
Required probability $=\frac{53}{3 \times 5 \times 7}=\frac{53}{105}$