The value of displaystyle∑r=020 50- r C 6 is equal to :

Question

The value of displaystyle∑r=020 50- r C 6 is equal to : (A)  51 C 7+ 30 C 7 (B)  51 C 7- 30 C 7 (C)  50 C 7- 30 C 7 (D)  50 C 6- 30 C 6

Tardigrade Admin · Accepted Answer

displaystyle∑r=020 50- r C 6= 50 C 6+ 49 C 6+ 48 C 6+ ldots . .+ 30 C 6 = 50 C 6+ 49 C 6+ ldots . .+ 31 C 6+( 30 C 6+ 30 C 7)- 30 C 7 = 50 C 6+ 49 C 6+ ldots . .+( 31 C 6+ 31 C 7)- 30 C 7 = 50 C 6+ 50 C 7- 30 C 7 = 51 C 7- 30 C 7 n C r + n C r -1= n +1 C r

Q. The value of $r = 0 \sum 20^{50 - r} C_{6}$ is equal to :

$^{51} C_{7} +^{30} C_{7}$

$^{51} C_{7} -^{30} C_{7}$

$^{50} C_{7} -^{30} C_{7}$

$^{50} C_{6} -^{30} C_{6}$

Q. The value of r=0∑20​50−rC6​ is equal to :

51C7​+30C7​

51C7​−30C7​

50C7​−30C7​

50C6​−30C6​

r=0∑20​50−rC6​=50C6​+49C6​+48C6​+…..+30C6​ =50C6​+49C6​+…..+31C6​+(30C6​+30C7​)−30C7​ =50C6​+49C6​+…..+(31C6​+31C7​)−30C7​ =50C6​+50C7​−30C7​ =51C7​−30C7​ nCr​+nCr−1​=n+1Cr​

Q. The value of $r = 0 \sum 20^{50 - r} C_{6}$ is equal to :

$^{51} C_{7} +^{30} C_{7}$

$^{51} C_{7} -^{30} C_{7}$

$^{50} C_{7} -^{30} C_{7}$

$^{50} C_{6} -^{30} C_{6}$