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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Permutation and Combination

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Permutation and Combination

The number of ways of getting a sum 16 on throwing a dice four...

The number of ways of getting a sum 16 on throwing a dice four times is______

Correct answer

0

Step-by-step solution

(x1+x2…+x6)4x4⋅(1−x61−x)4x4⋅(1−x6)4⋅(1−x)−4x4[1−4x6+6x12….][(1−x)−4](x4−4x10+6x16…)(1−x)−4(x4−4x10+6x16)(1+15C12x12+9C6x6….)(15C12−4⋅9C6+6)x16(15C3−4⋅9C6+6)=35×13−6×8×7+6=455−336+6=125\begin{aligned} & \left(x^1+x^2 \ldots+x^6\right)^4 \\ & x^4 \cdot\left(\frac{1-x^6}{1-x}\right)^4 \\ & x^4 \cdot\left(1-x^6\right)^4 \cdot(1-x)^{-4} \\ & x^4\left[1-4 x^6+6 x^{12} \ldots .\right]\left[(1-x)^{-4}\right] \\ & \left(x^4-4 x^{10}+6 x^{16} \ldots\right)(1-x)^{-4} \\ & \left(x^4-4 x^{10}+6 x^{16}\right)\left(1+{ }^{15} C_{12} x^{12}+{ }^9 C_6 x^6 \ldots .\right) \\ & \left({ }^{15} C_{12}-4 \cdot{ }^9 C_6+6\right) x^{16} \\ & \left({ }^{15} C_3-4 \cdot{ }^9 C_6+6\right) \\ & =35 \times 13-6 \times 8 \times 7+6 \\ & =455-336+6 \\ & =125\end{aligned}
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