12)the no of "words"of 4 letters consisting of equal number of vowels and consonants of the english language(repetition of letters allowed)
if (1+x+3x^2)^{30}=a_0+a_1x+a_2x^2............a_{60}x^{60}
then find the value of a_{5}+a_{7}+a_{9}+.........a_{59}
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4 Answers
1
1Put x=1 : a0 + a1 + a2 + a3 + ..... + a60 = 530
Put x = -1 : a0 - a1 + a2 - a3 + ..... + a60 = 330
Subtracting : 2(a1 + a3 + a5 + ..... + a59) = 530 - 330
a_{1}=\frac{d}{dx}(1+x+3x^2)^{30}|_{x=0}
6a_{3}=\frac{d^3}{dx^3}(1+x+3x^2)^{30}|_{x=0}