binomial theorem

(a) Itis given that [(1+3x)^m]+(1+5x)^n=2+19x+ax^2+terms involving higher powers
of x. If m and n are positive integers,find the values ofm and n.
Sol
(1+3x)^n=1+C(m，1)*(3x)+C(m，2)*(3x)^2+…
(1+5x)^n=1+C(n，1)*(5x)+C(n，2)*(5x)^2+…
So
3*C(m，1)+5*C(n，1)=19
3m+5n=19
[1] n=1
3m=14(reject)
[2] n=2
3m=9
m=3
a=9*C(3，2)+25*C(2，2)=27+25=52
[3] n=3
3m=4(reject)
(b) Hence find the value of a.
a=52

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