Question

A batsman can score $0,1,2,3,4$ or $6$ runs from a ball. The number of different sequences in which he can score exactly $30$ runs in an over of six balls is:

• A. $4$
• B. $72$
• C. $56$
• D. $71$

Answer 1

The correct option is D $71$

Possible cases of scoring $30$ runs:

Case I: $\{6,6,6,6,6,0\}$

Total sequence: $\frac{6!}{5!}=6$

Case II: $\{6,6,6,6,4,2\}$

Total sequence: $\frac{6!}{4!}=30$

Case III: $\{6,6,6,6,3,3\}$

Total sequence: $\frac{6!}{2!4!}=15$

Case IV: $\{6,6,6,4,4,4\}$

Total sequence: $\frac{6!}{3!3!}=20$

Hence total number of sequence is: $6+30+15+20=71$