CONCEPT: Binomial Theorem: According to the theorem, it is possible to expand any non-negative power of x+ y into a sum of the form: (x+y)n=nC0xny0+nC1xn−1y1+nC2xn−2y2+nC3xn−3y3+......+nCnx0yn where n≥0 is an integer. CALCULATION: (1+x)50=(1+50C1x1+50C2x2+50C3x3+50C4x4+......+50C50x50) ......equation (1) The above equations is written with the help of “BINOMIAL THEOREM” We need to find the sum of left(50C1+50C3+50C5+.......+50C49) , as they are the coefficients of odd powers of x . In the equation (1), substitute x = 1 ⇒(1+1)50=1+50C1+50C2+50C3+50C4+......+50C50⇒250=1+50C1+50C2+50C3+50C4+......+50C50...... equation (2) In the equation(1), substitute x = (-1) ⇒(1−1)50=1−50C1+50C2−50C3+50C4+̣.....+50C50⇒0=1−50C1+50C2−50C3+50C4+......+50C50......equation (3) By subtracting equation (3) from equation (2), ⇒250=2×(50C1+50C3+50C5+50C7+......+50C49⇒50C1+50C3+50C5+50C7+.......+50C49=249 The sum of the coefficients of odd powers of x=249