Prove that: C r n - 1 + C r n - 2 + . . . + C r r = C r + 1 n .

LHS = C r r + C r r + 1 + C r r + 2 + … . . + C r n - 2 + C r n - 1 = C r + 1 r + 1 + C r r + 1 + C r r + 2 + … . + C r n - 2 + C r n - 1 ∵ C r r = C r + 1 r + 1 = C r + 1 r + 2 + C r + 2 r + C r + 3 r + … + C r n - 2 + C r n - 1 ∵ C r n + C r - 1 n = C r n + 1 = C r + 1 r + 3 + C r + 3 r + C r + 4 r + … . . . + C r n - 2 + C r n - 1 Adding first two terms each time, at last we get, = C r + 1 n - 1 + C n - 1 r = C n r = RHS Hence, proved.

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Let $8$ persons can be seated on a round table, then the number of ways

(a) are $M$ , if two of them (say $A$ and $B$ ) must not sit in adjacent seats.

(b) are $N$ , if $4$ of the persons are men and $4$ ladies and if no two men are to be in adjacent seats.

(c) are $P$ , if $8$ persons constitute $4$ married couples and if no husband and wife, as well as no two men are to be in adjacent seats.

Find the value of $(\frac{M - N}{10 P})$ .

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Important Questions on Permutation and Combination

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 24

There are

2

women participating in a chess tournament. Every participant played

2

games with the other participants. The number of games that the men played between themselves exceeded by

66

as compared to the number of games that the men played with the women. If number of participants & the total number of games played in the tournament are

M

and

N

respectively, then value of

(M + N)

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 25

Let

15

identical candy bars be distributed between Ram, Shyam, Ghanshyam and Balram, if Ram can not have more than

5

candy bars and Shyam must have at least two are

N

, then sum of digits of

N

is (Assume all candy bars to be alike)

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 26

If the sum of all numbers greater than

10000

formed by using the digits

0, 1, 2, 4, 5

& no digit being repeated in any number, is

N

, then sum of digits of

N

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 27

Let the number of ways in which the letters of the word? 'MUNMUN' can be arranged so that no two alike letters are together are

N

, then value of

N

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 28

A shop sells $6$ different flavours of ice-creams. Let the number ways can a customer choose $4$ ice-cream cones

(a) are $M$ , if they are all different flavours.

(b) are $N$ , if they are not necessarily of different flavours.

(d) are $Q$ , if they contain only $2$ or $3$ different flavours.

Find the value of $(N + P) - (M + Q)$ .

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 29

If number of ways in which a selection of

100

balls, can be made out of

100

identical red balls,

100

identical blue balls &

100

identical white balls is

N

, then sum of digits of

N

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-3>Q 30

There are

5

balls of different colours &

5

boxes of colours same as those of the balls. The number of ways in which the balls, one in each box could be placed such that exactly one ball goes to the box of its own colour are

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Beta Question Bank for Engineering: Mathematics>Chapter 9 - Permutation and Combination>EXERCISE-4>Q 1

Prove that:

{}^{n - 1}C_{r} + {}^{n - 2}C_{r} + . . . + {}^{r}C_{r} = {}^{n}C_{r + 1}

Important Questions on Permutation and Combination

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