Chapter 11.1, Problem 38SR

To determine

Find the sequences of the hits and misses.

Answer to Problem 38SR

  $45$

Explanation of Solution

Given information:

Ten shots in the intramural free- throw shooting competition.

Calculation:

Let the number of hits by Ofelia is taken as $h$ and number of misses as $m$ .

Now the expression for ten shots taken by Ofelia is ${\left(h+m\right)}^{10}$

Apply binomial expansion for ${\left(h+m\right)}^{10}$ expression, we get.

  ${(h+m)}^{10}{=}^{10}{C}_0{h}^{10}{m}^0{+}^{10}{C}_1{h}^9{m}^1{+}^{10}{C}_2{h}^5{m}^2{+}^{10}{C}_3{h}^7{m}^3{+}^{10}{C}_4{h}^6{m}^4$

  ${+}^{10}{C}_3{h}^5{m}^5{+}^{10}{C}_6{h}^4{m}^6{+}^{10}{C}_7{h}^3{m}^7{+}^{10}{C}_3{h}^2{m}^8{+}^{10}{C}_{9}h{m}^9{+}^{10}{C}_{10}{h}^0{m}^{10}$

  $=\frac{10!}{0!(10-0)!}{h}^0{m}^0+\frac{10!}{1!(10-1)!}{h}^9{m}^1+\frac{10!}{2!(10-2)!}{h}^8{m}^2+\frac{10!}{3!(10-3)!}{h}^7{m}^3+\frac{10!}{4!(10-4)!}{h}^6{m}^4$

  $+\frac{10!}{5!(10-5)!}{h}^5{m}^5+\frac{10!}{6!(10-6)!}{h}^4{m}^6+\frac{10!}{7!(10-7)!}{h}^3{m}^7+\frac{10!}{8!(10-8)!}{h}^2{m}^8+\frac{10!}{9!(10-9)!}{h}^4{m}^9+\frac{10!}{10!(10-10)!}{h}^0{m}^{10}$

  $=\frac{10!}{0!10!}{h}^{10}{m}^0+\frac{10!}{1!9!}{h}^9{m}^1+\frac{10!}{2!8!}{h}^5{m}^2+\frac{10!}{3!7!}{h}^7{m}^3+\frac{10!}{4!6!}{h}^6{m}^4$

  $+\frac{10!}{5!5!}{h}^5{m}^3+\frac{10!}{6!4!}{h}^4{m}^6+\frac{10!}{7!3!}{h}^3{m}^7+\frac{10!}{8!2!}{h}^2{m}^8+\frac{10!}{9!1!}{h}^1{m}^9+\frac{10!}{10!0!}{h}^0{m}^{10}$

  $=h^{10}{m}^0+\frac{10.9!}{119!}{h}^9{m}^1+\frac{\mathrm{10.9.8}!}{2!8!}{h}^8{m}^2+\frac{\mathrm{10.9.8.7}!}{3!7!}{h}^7{m}^3+\frac{\mathrm{10.9.8.7.6}!}{4!6!}{h}^6{m}^4$

  $+\frac{\mathrm{10.9.8.7.6.5}!}{5!5!}{h}^5{m}^5+\frac{\mathrm{10.9.8.7.6}!}{6!4!}{h}^4{m}^6+\frac{\mathrm{10.9.8.7}!}{7!3!}{h}^3{m}^7+\frac{\mathrm{10.9.8}!}{8!2!}{h}^2{m}^8+\frac{10.9!}{9!1!}{h}^{\prime }{m}^9+h^0{m}^{10}$

  $=h^{10}{m}^0+10h^9{m}^1+45h^5{m}^2+120h^7{m}^3+210h^6{m}^4$

  $+252h^5{m}^5+210h^4{m}^6+120h^3{m}^7+45h^2{m}^8+10h^1{m}^9+h^0{m}^{10}$

The term $45h^8{m}^2$ satisfy the condition of making eight shots and two shots missing.

Hence, the sequences of the hits and misses are there that result in her making eight shots and missing two is $45$ .