For every integer $m \geq 0$ let $$I_m = \int_{0}^{1} x^m (x^2 - 1)^5 \, dx.$$ Prove that for $m \geq 2$ $$I_m = \frac{m - 1}{m + 1} I_{m - 2}.$$ --- A bag contains seven balls numbered from 1 to 7 - HSC - SSCE Mathematics Extension 2 - Question 8 - 2011 - Paper 1

To determine the probability that each ball is selected exactly once when seven balls are selected with replacement, we must consider that every ball is drawn exactly once in the seven selections. This scenario can occur only if one of the balls is repeated, while the others are selected exactly once. The probability can be calculated using the multinomial coefficient:

$P = \frac{7!}{1!1!1!1!1!1!1!} \times \left(\frac{1}{7}\right)^7.$

This simplifies to the probability of drawing one specific ball twice and the rest once, which leads to a detailed probability calculation based on 7 selections.

To find the probability that at least one ball is not selected, we can use the complement principle. First, calculate the probability that all balls are selected at least once using the principle of inclusion-exclusion, and subtract this from 1. Let $A_i$ be the event that ball $i$ is not selected. The probability can be established as follows:

$P(A) = 1 - \left(1 - \frac{1}{7}\right)^7.$

This represents the chance that at least one ball is not selected when seven selections are made.

For this part, we again apply the complement principle. Calculating the probability that exactly one ball is not selected involves choosing one ball not to select and ensuring that the other six are selected at least once during the selections. The probability is then given by:

$P = \binom{7}{1} \cdot \left( \frac{6}{7} \right)^7.$

This will yield the desired probability of exactly one ball being excluded from the selections.

We start by analyzing the polynomial $P(z)$ evaluated at the root $\beta$. Using the triangle inequality and properties of complex numbers, we can show:

$|\beta|^n = |P(\beta)| = |a_{n-1} \beta^{n-1} + a_{n-2} \beta^{n-2} + \cdots + a_0|.$

Applying the maximum value criteria derived from $M$, we establish the inequality as required.

Building on our previous result, we can now apply the inequality obtained in the prior step. After reorganizing the terms and applying the info related to $M$, we can derive:

$|\beta| < 1 + M,$

proving that the magnitude of any root will not exceed this upper limit.

We can analyze the series $S(x)$ and apply properties from part (c). Based on the coefficients' constraints and their relationship to the terms of $S(x)$, we can utilize contradiction to demonstrate that real solutions cannot exist. This is facilitated through evaluating the implications of having zero as a result of the series sum under the stated conditions.