Mathematics - Binomial Theorem Question with Solution | TestHub

The $(r+1{)}^{th}$ term is given by $T_{r+1}={}^{21}{\mathrm{C}}_r{a}^{\frac{42-3r}{6}}{b}^{\frac{2r-21}{3}}$.

To find the value of $r$, we set the exponents of $a$ and $b$ equal:

$$\frac{42-3r}{6}=\frac{2r-21}{3}$$

Multiplying both sides by 6 to clear the denominators, we get:

$$42-3r=2(2r-21)$$

$$42-3r=4r-42$$

Now, we solve for $r$:

$$42+42=4r+3r$$

$$84=7r$$

$$r=\frac{84}{7}$$

$$r=12$$