Question

after a genetics experiment on 63 pea plants, the number of plants having certain characteristics was tallied, with the results given to the right. answer parts (a) through (c). (a) find the number of plants that were tall and had smooth peas. 31 were tall; 31 had green peas; 44 had smooth peas; 13 were tall and had green peas; 22 had green peas and smooth peas; 7 had all three characteristics; 3 had none of the characteristics.

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(T\) be the set of tall plants, \(G\) be the set of plants with green peas, and \(S\) be the set of plants with smooth peas. The total number of non - empty plants is \(n = 63-3=60\).
The formula for \(n(T\cup G\cup S)\) is \(n(T)+n(G)+n(S)-n(T\cap G)-n(G\cap S)-n(T\cap S)+n(T\cap G\cap S)\).
We know \(n(T) = 31\), \(n(G)=31\), \(n(S)=44\), \(n(T\cap G)=13\), \(n(G\cap S)=22\), \(n(T\cap G\cap S)=7\), and \(n(T\cup G\cup S)=60\).

Step2: Substitute into the formula and solve for \(n(T\cap S)\)

\[

$$\begin{align*} n(T\cup G\cup S)&=n(T)+n(G)+n(S)-n(T\cap G)-n(G\cap S)-n(T\cap S)+n(T\cap G\cap S)\\ 60&=31 + 31+44-13-22 - n(T\cap S)+7\\ 60&=(31 + 31+44+7)-(13 + 22)-n(T\cap S)\\ 60&=113 - 35 - n(T\cap S)\\ 60&=78 - n(T\cap S)\\ n(T\cap S)&=78 - 60\\ n(T\cap S)&=18 \end{align*}$$

\]

Answer:

18