Chapter 5

Prove by combinatorial argument that \({ }^{n+1} \mathrm{C}_{r}={ }^{n} \mathrm{C}_{r}+{ }^{n} \mathrm{C}_{r-1}\).

If \(\alpha={ }^{m} C_{2}\), then \({ }^{\alpha} C_{2}\) is equal to a. \({ }^{m+1} C_{4}\) b. \({ }^{m-1} C_{4}\) c. \(3^{m+7} C_{i}\) d. \(3^{m+1} C_{4}\)

Number of ways in which three numbers in A.P. can be selected from \(1,2,3, \ldots, n\) is a. \(\left(\frac{n-1}{2}\right)^{2}\) if \(n\) is even b. \(\frac{n(n-2)}{4}\) if \(n\) is even c. \(\frac{(n-1)^{2}}{4}\) if \(n\) is odd d. none of these

Prove that \((n !) !\) is divisible by \((n !)^{(n-1) !}\).

If \({ }^{\prime} \mathrm{C}_{3}+{ }^{n} \mathrm{C}_{4}>{ }^{n+1} C_{3}\), then a. \(n>6\) b. \(n>7\) c. \(n<6\) d. none of these

Kanchan has 10 friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is a. \({ }^{8} C_{5}\) b. \(2 \times{ }^{8} C_{3}\) c. \({ }^{10} C_{5}-2 \times{ }^{8} C_{4}\) d. none of these

If \(n_{1}\) and \(n_{2}\) are five-digit numbers, find the total number of ways of forming \(n_{1}\) and \(n_{2}\) so that these numbers can be added without carrying at any stage.

The value of \(\sum_{r=0}^{n-1}{\underline{\phantom{xx}}}^{n} C_{r} /\left({ }^{n} C_{r}+{ }^{n} C_{r+1}\right)\) equals a. \(n+1\) b. \(n / 2\) c. \(n+2\) d. none of these

A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let, \(p=\) number of forecasts with exactly I error \(q=\) number of forecasts with exactly 3 errors and \(r=\) number of forecasts with all five errors Then the correct statement(s) is/are a. \(2 q=5 r\) b. \(8 p=q\) c. \(8 p=5 r\) d. \(2(p+r)>q\)

\(n_{1}\) and \(n_{2}\) are four-digit numbers. Find the total number of ways of forming \(n_{1}\) and \(n_{2}\) so that \(n_{2}\) can be subtracted from \(n_{1}\) without borrowing at any stage.

In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, the maximum population of the city is a. \(2^{32}\) b. \((32)^{2}-1\) c. \(2^{32}-1\) d. \(2^{32-1}\)

Ten persons numbered \(1,2, \ldots, 10\) play a chess tournament, each player playing against every other player exactly one game. It is known that no game ends in a draw. If \(w_{1}, w_{2}, \ldots, w_{10}\) are the number of games won by players \(1,2,3, \ldots, 10\), respectively, and \(I_{1}, l_{2}, \ldots, l_{10}\) are the number of games lost by the players \(1,2, \ldots\) 1 \(O\), respectively, then, a. \(\sum w_{l}=\Sigma l_{i}=45\) b. \(w_{i}+1_{i}=9\) c. \(\sum w l_{1}^{2}=81+\Sigma l_{1}^{2}\) d. \(\sum w_{i}^{2}=\sum l_{i}^{2}\)

How many five-digit numbers can be made having exactly two identical digits?

In a room, there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is a. \(12^{2}-1\) b. \(2^{12}\) c. \(2^{12}-1\) d. \(12^{2}\)

The number of ways of choosing triplet \((x, y, z)\) such that \(z \geq \max \\{x, y\\}\) and \(x, y, z \in\\{1,2, \ldots, n, n+1\\}\) is a. \({ }^{n+1} C_{3}+{ }^{n+2} C_{3}\) b. \(n(n+1)(2 n+1) / 6\) c. \(1^{2}+2^{2}+\cdots+n^{2}\) d. \(2\left({ }^{(n+2} C_{3}\right)-{ }^{n+1} C_{2}\)

An ordinary cubical dice having six faces marked with alphabets \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and \(\mathrm{F}\) is thrown \(n\) times and the list of \(n\) alphabets showing up are noted. Find the total number of ways in which among the alphabets \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and \(\mathrm{F}\) only three of them appear in the list.

The number of possible outcomes in a throw of \(n\) ordinary dice in which at least one of the dice shows an odd number is a. \(6^{n}-1\) b. \(3^{n}-1\) c. \(6^{n}-3^{n}\) d. none of these

Number of ways in which 200 people can be divided in 100 couples is a. \(\frac{(200) !}{2^{100}(100) !}\) b. \(1 \times 3 \times 5 \cdots 199\) c. \(\left(\frac{101}{2}\right)\left(\frac{102}{2}\right) \cdots\left(\frac{200}{2}\right)\) d. \(\frac{(200) !}{(100) !}\)

Find the number of three-digit numbers from 100 to 999 including all numbers which have any one digit that is the average of the other two.

Let \(A\) be a set of \(n(\geq 3)\) distinct elements. The number of triplets \((x, y, z)\) of the \(A\) elements in which at least two coordinates is equal to a. \({ }^{w} \mathrm{P}_{3}\) b. \(n^{3}-{ }^{n} \mathrm{P}_{3}\) c. \(3 n^{2}-2 n\) d. \(3 n^{2}(n-1)\)

If a seven-digit number made up of all distinct digits \(8,7,6,4,3\) \(x\) and \(y\) is divisible by 3 , then a. maximum value of \(x-y\) is 9 b. maximum value of \(x+y\) is 12 c. minimum value of \(x y\) is 0 d. minimum value of \(x+y\) is 3

The total number of flags with three horizontal strips in order, which can be formed using 2 identical red, 2 identical green and 2 identical white strips is equal to a. \(4 !\) b. \(3 \times(4 !)\) c. \(2 \times(4 !)\) d. none of these

If \(n\) is number of necklaces which can be formed using 17 identical pearls and two identical diamonds and similarly \(m\) is number of necklaces which can be formed using 17 identical pearls and different diamonds, then a. \(n=9\) b. \(m=18\) c. \(n=18\) d. \(m=9\)

There are \(2 n\) guests at a dinner party. Supposing that the master and mistress of the house have fixed seats opposite one another and that there are two specified guests who must not be placed next to one another, show that the number of ways in which the company can be placed is \((2 n-2) ! \times\left(4 n^{2}-6 n+4\right)\).

The number of five-digit numbers that contain 7 exactly once is a. \((41)\left(9^{3}\right)\) b. \((37)\left(9^{3}\right)\) c. (7) \(\left(9^{4}\right)\) d. \((41)\left(9^{4}\right)\)

Let \(f(n)\) be the number of regions in which \(n\) coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different point and no three of them have common point of intersection, then a. \(f(20)=382\) b. \(f(n)\) is always an even number c. \(f^{-1}(92)=10\) d. \(f(n)\) can be odd

In how many ways can two distinct subsets of the set \(A\) of \(k\) \((k \geq 2)\) elements be selected so that they have exactly two common elements?

A variable name in certain computer language must be eithér an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to a. 280 b. 290 c. 286 d. 296

Given that the divisors of \(n=3^{p} \cdot 5^{q} \cdot 7^{\prime}\) are of the form \(4 \lambda+1\), \(\lambda \geq 0\). Then a. \(p+r\) is always even b. \(p+q+r\) is always odd c. \(q\) can be any integer d. if \(p\) is odd then \(r\) is even

Prove that the number of ways to select \(n\) objects from \(3 n\) objects of which \(n\) are identical and the rest are different is $$ 2^{2 n-1}+\frac{1}{2} \frac{2 n !}{(n !)^{2}} $$

The number less than 1000 that can be formed using the digits 0 , \(1,2,3,4,5\) when repetition is not allowed is equal to a. 130 b. 131 c. 156 d. 155

Number of ways of selecting three integers from \(\\{1,2,3, \ldots, n\\}\) if their sum is divisible by 3 is a. \(3\left({ }^{n / 3} C_{3}\right)+(n / 3)^{3}\) if \(n=3 k, k \in N\) b. \(2\left({ }^{\left(1 n-10^{\circ}\right.} C_{3}\right)+\left({ }^{(n+2) 3} C_{3}\right)+((n-1) / 3)^{2}(n+2)\), if \(n=3 k+1\), \(k \in N\) c. \(2\left({ }^{(n-1) \times 3} C_{3}\right)+\left({ }^{(n+2) 3} C_{3}\right)+((n-1) / 3)^{2}(n+2)\), if \(n=3 k+2\), \(k \in N\) d. independent of \(n\)

There are \(n\) straight lines in a plane, in which no two are parallel and no three pass through the same point. Their points of intersection are joined. Show that the number of fresh lines thus introduced is $$ \frac{1}{8} n(n-1)(n-2)(n-3) $$

Total number of six-digit numbers that can be formed, having the property that every succeeding digit is greater than the preceding digit, is equal to a. \({ }^{9} \mathrm{C}_{3}\) b. \({ }^{10} C_{3}\) c. \({ }^{9} p_{3}\) d. \({ }^{10} p_{3}\)

Number of points of intersection of \(n\) straight lines if \(n\) satisfies \({ }^{n+5} P_{n+1}=\frac{11(n-1)}{2} \times{ }^{n+3} P_{n}\) is a. 15 b. 28 c. 21 d. 10

There are \(n\) points in a plane, in which no three are in a straight line except ' \(m\) ' which are all in a straight line. Find the number of (a) different straight lines, (b) different triangles, (c) different quadrilaterals that can be formed with the given points as vertices.

Numbers greater than 1000 but not greater than 4000 , which can be formed with the digits \(0,1,2,3,4\) (repetition of digits is allowed), are a. 350 b. 375 c. 450 d. 576

Number of shortest ways in which we can reach from the point \((0,0,0)\) to point \((3,7,11)\) in space where the movement is possible only along the \(x\)-axis, \(y\)-axis and z-axis or parallel to them and change of axes is permitted only at integral points (an integral point is one which has its coordinate as integer) is a. equivalent to number of ways of dividing 21 different objects in three groups of size \(3,7,11\) b. equivalent to coefficient of \(y^{3} z^{7}\) in the expansion of \((1+y+z)^{21}\) c. equivalent to number of ways of distributing 21 different objects in three boxes of size \(3,7,11\) d. equivalent to number of ways of arranging 21 objects of which 3 are alike of one kind, 7 are alike of second type and 11 are alike of third type

Find the number of ways of disturbing \(n\) identical objects among \(n\) persons if at least \(n-3\) persons get none of these objects.

The total number of five-digit numbers of different digits in which the digit in the middle is the largest is a. \(\sum_{n=4}^{9} " P_{4}\) b. \(33(3 !)\) c. \(30(3 !)\) d. none of these

If \(N\) denotes the number of ways of selecting \(r\) objects out of \(n\) distinct objects \((r \geq n)\) with unlimited repetition but with each object included at least once in selection, then \(N\) is equal to a. \(\quad{ }^{r-1} C_{r-n}\) b. \({ }^{r-1} C_{n}\) c. \({ }^{r-1} C_{n-1}\) d. none of these

A batsman scores exactly a century by hitting fours and sixes in twenty consecutive balls. In how many different ways can he do it if some balls may not yield runs and the order of boundaries and overboundaries are taken into account?

In how many ways can \(2 t+1\) identical balls be placed in three distinct boxes so that any two boxes together will contain more balls than the third?

Total number of words that can be formed using all letters of the word 'BRIJESH' that neither begins with 'I' nor ends with ' \(\mathrm{B}^{\prime}\) is equal to a. 3720 b. 4920 c. 3600 d. 4800

Sohan has \(x\) children by his first wife. Geeta has \((x+1)\) children by her first husband. They marry and have children of their own. The whole family has 24 children. Assuming that two children of the same parents do not fight, prove that the maximum possible number of fights that can take place is 191 .

The total number of six-digit natural numbers that can be made with the digits \(1,2,3,4\), if all digits are to appear in the same number at least once is a. 1560 b. 840 c. 1080 d. 480

If \(P=21\left(21^{2}-1^{2}\right)\left(21^{2}-2^{2}\right)\left(21^{2}-3^{2}\right) \cdots\left(21^{2}-10^{2}\right)\), then \(P\) is divisible by a. \(22 !\) b. \(21 !\) c. \(19 !\) d. \(20 !\)

Let \(S(n)\) denote the number of ordered pairs \((x, y)\) satisfying \(1 / x\) \(+1 / y=1 / n\) where \(n>1\) and \(x, y, n \in N\) (i) Find the value of \(S(6)\). (ii) Show that if \(n\) is prime, then \(S(n)=3\) always.

Total number of six-digit numbers in which all and only odd digits appear is a. \(\frac{5}{2}(6 !)\) b. \(6 !\) c. \(\frac{1}{2}(6 !)\) d. none of these

Six apples and six mangoes are to be distributed among ten boys so that each boy receives at least one fruit. Find the number of ways in which the fruits can be distributed.