52 different cards (of which 4 are aces) are distributed equally among 4 players. Shew that in nearly three-fifths of the possible distributions one player will have two aces and two other players one ace each.
Solution: This is \begin{align*} && P &= \frac{\binom{4}{2}\binom{48}{11}}{\binom{52}{13}} \cdot \frac{\binom{2}{1}\binom{37}{12}}{\binom{39}{13}}\cdot\frac{\binom{1}{1}\binom{25}{12}}{\binom{26}{13}} \cdot 4 \cdot 3 \\ &&&= \frac{6 \cdot 48! \cdot 13! \cdot 39!}{52! \cdot 11! \cdot 37!} \cdot \frac{2 \cdot 37! \cdot 13! \cdot 26!}{39! \cdot 12! \cdot 25!} \cdot \frac{25! \cdot 13! \cdot 13!}{26! \cdot 12! \cdot 13!} \cdot 4 \cdot 3 \\ &&&= \frac{6 \cdot 13 \cdot 12 }{52 \cdot 51 \cdot 50 \cdot 49 } \cdot 26 \cdot 13 \cdot 4 \cdot 3 \\ &&&= \frac{6 \cdot 13 \cdot 12 }{17\cdot 25\cdot 49 } \cdot 13\\ &&&= \frac{3}{5} \frac{2 \cdot 12 \cdot 13\cdot 13}{5 \cdot 17 \cdot 49}\\ &&&= \frac35 \cdot \frac{4056}{4165} \\ &&&\approx \frac35 \end{align*}
Prove that \[ \frac{u_1}{1-} \frac{u_1u_2}{u_1+u_2-} \frac{u_2u_3}{u_2+u_3-} \dots \frac{u_{n-1}u_n}{u_{n-1}+u_n} = u_1+u_2+u_3+\dots+u_n. \] Prove that \[ e^x = 1 + \frac{x}{1-x+} \frac{x}{2-x+} \frac{2x}{3-x+} \frac{3x}{4- \dots}. \]
An equilateral triangle is constructed with its angular points on the sides respectively of the triangle \(ABC\). Shew that the ratio of the area of the least equilateral triangle that can be so described to the area of \(ABC\) is \[ \left\{2+\frac{1}{\sqrt{3}}(\cot A + \cot B + \cot C)\right\}^{-1}. \]
The ordinates of three points \(P, Q, R\) on the parabola \(y^2=4ax\) are \(2al, 2am, 2an\). Shew that the area of the triangle \(PQR\) is \(a^2(m \sim n)(n \sim l)(l \sim m)\), and that the square of the radius of the circumscribing circle is \[ \frac{1}{16}a^2\{(m+n)^2+4\}\{(n+l)^2+4\}\{(l+m)^2+4\}. \]
A family of hyperbolas are drawn through the angular points of a triangle and the centre of its inscribed circle. Prove that the locus of their centres is a conic passing through the middle points of the sides of the triangle and through the points where these sides are cut by the bisectors of the opposite angles.
Forces \(P, -Q, R, -S\) act along the sides of a quadrilateral taken in order. Prove that they will be in equilibrium if \(P, Q, R, S\) are inversely proportional to the products of the lengths of the sides along which they act and the perpendiculars on those sides from the other two corners of the quadrilateral.
A body \(C\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, the coefficient of friction being \(\mu\), where \(\mu < \tan\alpha\). \(CP\) is drawn in the plane to represent a force just sufficient to maintain equilibrium. Prove that the locus of \(P\) is a circle, and that the greatest angle which \(CP\) can make with the line of greatest slope in the plane is \(\sin^{-1}(\mu\cot\alpha)\).
The range of a rifle bullet is 1200 yards when \(\alpha\) is the elevation of projection. Shew that if the rifle is fired with the same elevation from a car travelling at 10 miles per hour towards the target the range will be increased by \(220\tan\alpha\) feet.
A smooth straight tube \(BAC\) is bent at \(A\) and is fixed in a vertical plane so that \(AB, AC\) make angles \(\alpha, \beta\) on opposite sides of the vertical. A heavy uniform string \(PAQ\) in the tube is slightly displaced from the position of equilibrium; shew that when a length \(x\) has passed over \(A\) in the direction of \(P\) both the velocity and the acceleration vary as \(x\), and that the tension of the string at \(A\) varies as \((p+x)(q-x)\), where \(p, q\) are the lengths \(AP, AQ\) in the position of equilibrium. Shew also that the resultant vertical pressure on the tube is \(W\left(1 - \frac{pq}{l^2}\cos\alpha\cos\beta\right)\), where \(W\) is the weight of the string.
Solve the equations: