The number of rational terms in the binomial expansion of ( [4] 5 + [5] 4 )¹⁰⁰ is

Step-by-step solution

We have, aligned ( [4] 5 + [5] 4 )¹⁰⁰ & = _ r=0 ¹⁰⁰ ¹⁰⁰ C_r( [4] 5 )^ 100-r ( [5] 4 )^r & = _ r=0 ¹⁰⁰ ¹⁰⁰ C_r 5 100-r 4 4^ r 5 = _ r=0 ¹⁰⁰ T_ r+1 T_ r+1 & = ¹⁰⁰ C_r 5 100-r 4 4^ r 5 aligned Where, T_ r+1 = ¹⁰⁰ C_r 5 100-r 4 4^ r 5 Clearly, T_ r+1 will be an integer if 100-r 4 and r 5 are integers. This is possible when 100-r is a multiple of 4 and r is a multiple of 5 100-r=0,4,8,12, , 96,100 and r=0,5,10, , 100 r=0,4,8,12, , 100 and r=0,5,10,100 r=0,20,40,60,80,100 Hence, there are 6 rational terms.