displaystyle∑k=06 51-k C3 is equal to

Question

displaystyle∑k=06 51-k C3 is equal to (A)  51 C 3- 45 C 3 (B)  52 C 4- 45 C 4 (C)  52 C 3- 45 C 3 (D)  51 C 4- 45 C 4

Tardigrade Admin · Accepted Answer

displaystyle∑ k =06 51- k C 3 = 51 C 3+ 50 C 3+ 49 C 3+ ldots+ 45 C 3 = 45 C 3+ 46 C 3+ ldots . .+ 51 C 3 = 45 C 4+ 45 C 3+ 46 C 3+ ldots . .+ 51 C 3- 45 C 4 ( n Cr+ n Cr-1= n+1 Cr) = 52 C 4- 45 C 4

Q. $k = 0 \sum 6^{51 - k} C_{3}$ is equal to

$^{51} C_{3} -^{45} C_{3}$

$^{52} C_{4} -^{45} C_{4}$

$^{52} C_{3} -^{45} C_{3}$

$^{51} C_{4} -^{45} C_{4}$

$k = 0 \sum 6^{51 - k} C_{3}$
$=^{51} C_{3} +^{50} C_{3} +^{49} C_{3} + \dots +^{45} C_{3}$
$=^{45} C_{3} +^{46} C_{3} + \dots .. +^{51} C_{3}$
$=^{45} C_{4} +^{45} C_{3} +^{46} C_{3} + \dots .. +^{51} C_{3} -^{45} C_{4}$
$(^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r})$
$=^{52} C_{4} -^{45} C_{4}$

Q. k=0∑6​51−kC3​ is equal to

51C3​−45C3​

52C4​−45C4​

52C3​−45C3​

51C4​−45C4​