I found a solution for this that uses multinomial however I tried to solve it in a different way and I don't know where I am going wrong. So I thought how about we fill each box with one ball first, and then put the remaining 3 in all possible boxes. This was my reasoning: First let's choose ...

Let $V$ be an $n$ -dimensional vector space and let $B, B^{\prime}$ be two ordered bases for $V .$ Prove $P_{B B^{\prime}}$ is the only matrix with the following property: $$ [v]_{B^{\prime}}=P_{B B^{\prime}}[v]_{B} $$ So somehow I have to show if $P_{B B^{\prime}}[v]_{B}$ = $Q_{B B^{\prime}}[v]...

My task is to write volume integral in 3 coordinate systems : Cartesian, cylindrical and spherical. This integral shows volume of intersection of 2 spheres, first with center at $(0, 0,-3)$ and radius $5$ and second with center at $(0,0, 3)$ and radius $\sqrt{13}$.I am able to do it for Cartesian...

Goal: To gain a better understanding of Euclidean space conformally compactified into a unit ball. Question: How can I visualise and mathematically describe Euclidean space conformally compactified into a unit ball? My attempt: Start with the conformally compactified disk model of Eucl...

Consider the following sequence : Let $a_1 = a_2 = 1.$ For integer $ n > 2 : $ $$a_n = \frac{a_{n-1}(a_{n-1} + 1)}{a_{n-2}}.$$ $$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$ $$T = ??$$ What is the value of $T$ ? Is there a closed form or integral for $T$? I get $$ T = 3.73205080..$$...