Advertisements
Advertisements
प्रश्न
Write \[\sum^m_{r = 0} \ ^{n + r}{}{C}_r\] in the simplified form.
Advertisements
उत्तर
We know:
\[ \because \ ^{n}{}{C}_0 = \ ^{n + 1}{}{C}_0 \]
\[ \therefore \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 1}{}{C}_0 + \ ^{n + 1}{}{C}_1 + \ ^{n + 2}{}{C}_2 + \ ^{n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m \]
\[Using \ ^{n}{}{C}_{r - 1} + \ ^{n}{}{C}_r = \ ^{n + 1}{}{C}_r : \]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 2}{}{C}_1 + \ ^{n + 2}{}{C}_2 + \ ^{n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m \]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + 3}{}{C}_2 + \ ^ {n + 3}{}{C}_3 + . . . + \ ^{n + m}{}{C}_m\]
\[ \Rightarrow \sum^m_{r = 0} \ ^{n + r}{}{C}_r = \ ^{n + m + 1}{}{C}_m\]
APPEARS IN
संबंधित प्रश्न
How many chords can be drawn through 21 points on a circle?
Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?
Compute:
(i)\[\frac{30!}{28!}\]
Compute:
From Goa to Bombay there are two roots; air, and sea. From Bombay to Delhi there are three routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there?
A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?
There are 5 books on Mathematics and 6 books on Physics in a book shop. In how many ways can a students buy : (i) a Mathematics book and a Physics book (ii) either a Mathematics book or a Physics book?
A team consists of 6 boys and 4 girls and other has 5 boys and 3 girls. How many single matches can be arranged between the two teams when a boy plays against a boy and a girl plays against a girl?
From among the 36 teachers in a college, one principal, one vice-principal and the teacher-incharge are to be appointed. In how many ways can this be done?
How many three-digit odd numbers are there?
How many different numbers of six digits each can be formed from the digits 4, 5, 6, 7, 8, 9 when repetition of digits is not allowed?
A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.
If nC4 , nC5 and nC6 are in A.P., then find n.
If α = mC2, then find the value of αC2.
In how many ways can a football team of 11 players be selected from 16 players? How many of these will
exclude 2 particular players?
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: exactly 3 girls?
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: at least 3 girls?
A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated?
There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.
If 20Cr = 20Cr + 4 , then rC3 is equal to
If 20Cr + 1 = 20Cr − 1 , then r is equal to
If mC1 = nC2 , then
If\[\ ^{( a^2 - a)}{}{C}_2 = \ ^{( a^2 - a)}{}{C}_4\] , then a =
5C1 + 5C2 + 5C3 + 5C4 +5C5 is equal to
There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers?
If C0 + C1 + C2 + ... + Cn = 256, then 2nC2 is equal to
The value of\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + . . . + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\] is
A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends if two of the friends will not attend the party together is
A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics Part II, unless Mathematics Part I is also borrowed. In how many ways can he choose the three books to be borrowed?
All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of such arrangements in which no two vowels are adjacent to each other is ______.
The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers of m points are taken on l1; n points on l2, k points on l3. The maximum number of triangles formed with vertices at these points are ______.
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from the lot.
In how many ways can a football team of 11 players be selected from 16 players? How many of them will exclude 2 particular players?
The total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two signs ‘–’ occur together is ______.
A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee ______.
There are 3 books on Mathematics, 4 on Physics and 5 on English. How many different collections can be made such that each collection consists of:
| C1 | C2 |
| (a) One book of each subject; | (i) 3968 |
| (b) At least one book of each subject: | (ii) 60 |
| (c) At least one book of English: | (iii) 3255 |
All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is ______.
The number of positive integers satisfying the inequality `""^(n+1)C_(n-2) - ""^(n+1)C_(n-1) ≤ 100` is ______.
The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4, (repetition of digits is not allowed) and are multiple of 3 is?
