IB Mathematics Analysis and Approaches Topic AHL…

A till contains 60 coins, each of value 5 cents, 10 cents or 20 cents, and their total value is 700 cents.

Let $x$ , $y$ and $z$ be the numbers of 5-cent, 10-cent and 20-cent coins.

(a) Show that the two conditions on the coins give infinitely many real solutions, and that they are the triples $(x,\,y,\,z) = (2t-20,\,80-3t,\,t)$ .

(b) Find the number of ways the 60 coins could be made up.

(c) Three of the 60 coins are drawn at random, without replacement. Show that the probability that they have three different values is $\frac{(2t-20)(80-3t)t}{34220}$ .

(d) This probability is found to be $0.230$ , correct to three significant figures. Find all possible values of $x$ , $y$ and $z$ .

Consider the $7$ -letter word "ALGEBRA".

(a) Find how many distinct $5$ -letter arrangements can be formed from the letters of "ALGEBRA" if no letter is repeated.

(b) Find how many distinct $5$ -letter arrangements can be formed if the arrangement must start with "A", excluding those that end with "E". No repetition of letters is assumed.

Let $p$ and $q$ be real numbers, not both zero, and let $f(x) = \frac{px+q}{(1-2x)(1+3x)}$ .

(a) Show that $f(x) = \frac{A}{1-2x} + \frac{B}{1+3x}$ , where $A = \frac{p+2q}{5}$ and $B = \frac{3q-p}{5}$ .

(b) Show that the coefficient of $x^{n}$ in the expansion of $f(x)$ in ascending powers of $x$ is $A\,2^{n} + B(-3)^{n}$ , and the values of $x$ for which the expansion is valid.

The $n$ people at a dinner are to be seated in a row of $n$ chairs, one person to each chair.

Exactly $k$ of the $n$ people belong to one family, where $k \ge 2$ and $n \ge 2k$ .

(a) Show that the number of seatings in which no two family members occupy adjacent chairs is $(n-k)! \times \frac{(n-k+1)!}{(n-2k+1)!}$ .

(b) that the proportion of all seatings having this property is $\frac{(n-k)!\,(n-k+1)!}{n!\,(n-2k+1)!}$ .

IB Mathematics Analysis and Approaches AHL 1.10—perms and Combs, Binomial with Negative and Fractional Indices Questionbank

IB Mathematics Analysis and Approaches AHL 1.10—perms and Combs, Binomial with Negative and Fractional Indices Questionbank

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IB Mathematics Analysis and Approaches AHL 1.10—perms and Combs, Binomial with Negative and Fractional Indices Questionbank

IB Mathematics Analysis and Approaches AHL 1.10—perms and Combs, Binomial with Negative and Fractional Indices Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 1.10—perms and Combs, Binomial with Negative and Fractional Indices Questionbank

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