The probability that a missile hits a target successfully is 0 . 75...

Let the number of missiles be $n$,

$P$ (of destroying the target) $\ge 0 . 95$

$C3 n \cdot {\left(\dfrac{3}{4}\right)}^{3} \cdot {\left(\dfrac{1}{4}\right)}^{n - 3} + C4 n {\left(\dfrac{3}{4}\right)}^{4}{\left(\dfrac{1}{4}\right)}^{n - 4} + \ldots . . + Cn n {\left(\dfrac{3}{4}\right)}^{n} \ge 0 . 95$

$1 - \left\{C0 n {\left(\dfrac{3}{4}\right)}^{0} \cdot {\left(\dfrac{1}{4}\right)}^{n} + C1 n {\left(\dfrac{3}{4}\right)}^{1} \cdot {\left(\dfrac{1}{4}\right)}^{n - 1} + C2 n {\left(\dfrac{3}{4}\right)}^{2}{\left(\dfrac{1}{4}\right)}^{n - 2}\right\} \ge 0 . 95$

$1 - \dfrac{95}{100} \ge \dfrac{1}{{4}^{n}} + \dfrac{3n}{{4}^{n}} + \dfrac{n\left(n - 1\right)}{2} \cdot \dfrac{9}{{4}^{n}}$

$\dfrac{{4}^{n}}{20} \ge \dfrac{2 + 6n + 9{n}^{2} - 9n}{2}$

$\dfrac{{2}^{2n - 1}}{5} \ge 9{n}^{2} - 3n + 2$

${2}^{2n - 1} \ge 5\left(9{n}^{2} - 3n + 2\right)$

$n = 3 ⇒ 32 \ge 5 \times 74$ (not true)

$n = 4 ⇒ 128 \ge 5 \times 134$ (not true)

$n = 5 ⇒ 512 \ge 5 \times 212$ (not true)

$n = 6 ⇒ 2048 \ge 5 \times 308$ (true)

So, $n = 6$.