Advertisements
Advertisements
प्रश्न
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
विकल्प
27 − 1
28 − 2
28 − 1
28
Advertisements
उत्तर
28 − 2
\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + \left( \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right) + \left( \ ^{7}{}{C}_3 + \ ^{7}{}{C}_4 \right) + \left( \ ^{7}{}{C}_4 + \ ^{7}{}{C}_5 \right) + \left( \ ^{7}{}{C}_5 + \ ^{7}{}{C}_6 \right) + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\]
\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_4 + 2 \times \ ^{7}{}{C}_5 + 2 \times \ ^{7}{}{C}_6 + 1\]
\[= 1 + 2 \times \ ^{7}{}{C}_1 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_3 + 2 \times \ ^{7}{}{C}_2 + 2 \times \ ^{7}{}{C}_6 + 1\]
\[= 2 + 2^2 \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 + \ ^{7}{}{C}_3 \right)\]
\[ = 2 + 2^2 \left( 7 + \frac{7}{2} \times 6 + \frac{7}{3} \times \frac{6}{2} \times 5 \right)\]
\[= 2 + 252 \]
\[ = 254 \]
\[ = 2^8 - 2\]
APPEARS IN
संबंधित प्रश्न
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.
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?
Compute:
(i)\[\frac{30!}{28!}\]
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 mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?
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?
Twelve students complete in a race. In how many ways first three prizes be given?
How many different five-digit number licence plates can be made if
first digit cannot be zero and the repetition of digits is not allowed,
Since the number has to be greater than 8000, the thousand's place can be filled by only two digits, i.e. 8 and 9.
Now, the hundred's place can be filled with the remaining 4 digits as the repetition of the digits is not allowed.
The ten's place can be filled with the remaining 3 digits.
The unit's place can be filled with the remaining 2 digits.
Total numbers that can be formed = `2xx4xx3xx2=48`
In how many ways can six persons be seated in a row?
How many 3-digit numbers are there, with distinct digits, with each digit odd?
Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?
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.
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?
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:
a particular student is included.
In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (ii) at least one boy and one girl?
Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king?
We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can the selection be made?
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.
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 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: atmost 3 girls?
A parallelogram is cut by two sets of m lines parallel to its sides. Find the number of parallelograms thus formed.
Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (ii) triangles can be formed by joining them?
If 20Cr = 20Cr + 4 , then rC3 is equal to
If 15C3r = 15Cr + 3 , then r is equal to
If nCr + nCr + 1 = n + 1Cx , then x =
Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to
The number of diagonals that can be drawn by joining the vertices of an octagon is
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
Ten students are to be selected for a project from a class of 30 students. There are 4 students who want to be together either in the project or not in the project. Find the number of possible selections.
In how many ways can the letters of the word 'IMAGE' be arranged so that the vowels should always occupy odd places?
In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men and 5 women?
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 ______.
In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.
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?
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed is ______.
