Find the number of ways in which a batsman can score 14 runs in 6 balls

Find the number of ways in which a batsman can score 14 runs in 6 balls not scoring more than 4 runs in any ball
Ans:¹⁹C₅-6¹⁴C₅+15⁹C₅

2 Answers

Coefficient of x^{14} in (1+x+x^2+x^3+x^4)^6
is the reqd number of ways.
Don't ask me to find this - coz I'm bound to err!

@karan...
see on every ball batsman can score 0run,1 run....6runs
and we have six balls
also total runs =14

so no. of ways = coeff of x¹⁴ in (x⁰+x¹+...+x⁶)(x⁰+x¹+...+x⁶)(x⁰+x¹+...+x⁶)(x⁰+x¹+...+x⁶)(x⁰+x¹+...+x⁶)(x⁰+x¹+...+x⁶)

=>no. of ways = coeff of x¹⁴ in (x⁰+x¹+...+x⁶)⁶

But max runs off each ball =4
so we cant have x⁵ and x⁶

=>no. of ways=coeff of x¹⁴ in (x⁰+x¹+...+x⁴)⁶

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