Step 1 :Given the values: \(n = 54\), \(p = 0.6\), and \(X = 36\).
Step 2 :Calculate the binomial probability \(P(X)\) using the formula: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\).
Step 3 :Substitute the given values into the formula to get: \(P(X=36) = C(54, 36) * (0.6^{36}) * ((1-0.6)^{54-36})\).
Step 4 :Compute the above expression to get the probability \(P(X=36)\) which is approximately \(\boxed{0.0687}\).
Step 5 :Next, check if the normal distribution can be used to approximate this probability. The rule of thumb is that the normal distribution can be used if \(np(1-p) \geq 10\).
Step 6 :Calculate the value of \(np(1-p)\) by substituting the given values: \(54 * 0.6 * (1-0.6)\).
Step 7 :The calculated value is approximately 12.96, which is greater than 10.
Step 8 :Therefore, the normal distribution can be used to approximate this probability.
Step 9 :Final Answer: The probability \(P(X=36)\) is approximately \(\boxed{0.0687}\). The normal distribution can be used to approximate this probability because \(np(1-p)\) is \(\boxed{12.96}\), which is greater than 10. Therefore, the answer is \(\boxed{\text{D. Yes, the normal distribution can be used because } np(1-p) \geq 10}\).