Question

The expresion

${}^{45}{C}_8$+${\sum }_{k=1}^7{}^{52-k}{C}_7$+${\sum }_{i=1}^5{}^{57-i}{C}_{50-i}$

• A. ${}^{55}{C}_7$
• B. ${}^{57}{C}_8$
• C. ${}^{57}{C}_7$
• D. None of these

Answer 1

The correct option is B ${}^{57}{C}_8$

${}^{45}{C}_8$+${\sum }_{k=1}^7{}^{52-k}{C}_7$+${\sum }_{i=1}^5{}^{57-i}{C}_{50-i}$
$\Rightarrow$${}^{45}{C}_8$+${}^{45}{C}_7$+${}^{46}{C}_7$+......+${}^{51}{C}_7$+${\sum }_{i=1}^5{}^{57-i}{C}_{57-i-(50-i)}$
$\Rightarrow$(${}^{45}{C}_8$+${}^{45}{C}_7$)+${}^{46}{C}_7$+......+${}^{51}{C}_7$+${\sum }_{i=1}^5{}^{57-i}{C}_{7)}$
$\Rightarrow$(${}^{46}{C}_8$+${}^{46}{C}_7$)+......+${}^{51}{C}_7$+${}^{52}{C}_7$+${}^{53}{C}_7$+........+${}^{56}{C}_7$
:
:
$\Rightarrow$ ${}^{56}{C}_8$+${}^{56}{C}_7$
$\Rightarrow$ ${}^{57}{C}_8$