Question

# In how many ways can we get a sum of at most 17 by throwing six distinct dice -

- 6940
- 7881
- 9604
- None of these

## The correct answer is: 9604

### x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} 17

When 1 x_{i} 6, i = 1, 2, 3, …..6

Let x_{7} be a variable such that

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} = 17

Clearly x_{7} 0 Required number of ways

= Coefficient of x^{17} in (x^{1} + x^{2} + ….. + x^{6})^{6}

(1 + x + x^{2} + …..)

= Coefficient of x^{11} in

= Coefficient of x^{11} in (1–^{ 6}C_{1} x^{6} + ^{6}C_{2} x^{12}……) (1 – x)^{–7}

= Coefficient of x^{11 }in (1 – x)^{–7} – ^{6}C_{1} × coefficient of x^{5} in (1 – x)^{–7}

= ^{11+7–1}C_{7–1} – ^{6}C_{1} × ^{7+5–1}C_{7–1}

= ^{17}C_{6} – 6 × ^{11}C_{6} = 9604.

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