In the binomial expansion of (1+x)¹⁵ , the coefficients of x ^ r and x ^ r +3 are equal. Then, r is

Step-by-step solution

array lc Given, & (1+ x )¹⁵ Now, & T _ r +1 = ¹⁵ C _ r x ^ r and & T _ ( r +3)+1 = ¹⁵ C _ r +3 x ^ r +3 array According to question coefficient of x ^ r = coefficient of x ^ r +3 aligned & 15 r = 15 r+3 & 15 ! r !(15-r) ! = 15 ! (r+3) !(12-r) ! & 1 (15-r)(14-r)(13-r) & (r+1)(r+2)(r+3)= 1 (r+3)(r+2)(r+1) & (r²+3 r+2 )(r+3)= (210-29 r+r² )(13-r) aligned array lc & r³+3 r²+2 r+3 r²+9 r+6 & =2930-377 r+13 r²-210 r+29 r²-r³ & 2 r³-36 r²+598 r-2924=0 & r³-18 r²+299 r-1462=0 & (r-6) (r²-12 r+227 )=0 r & and r²-12 r+227=0