Given: The word \[INVOLUTE\] Total number of letters \[=\text{ }8\] Total vowels are \[=\text{ }I,\text{ }O,\text{ }U,\text{ }E\] Total consonants \[=\text{ }N,\text{ }V,\text{ }L,\text{ }T\] So...
Here, it is clear that \[3\]things are already selected and we need to choose \[\left( r\text{ }-\text{ }3 \right)\] things from the remaining \[\left( n\text{ }-\text{ }3 \right)\] things. Let us...
Given: The word \[MONDAY\] Total letters \[=\text{ }6\] All letters are used but first letter is a vowel ? In the word \[MONDAY\] the vowels are \[O\text{ }and\text{ }A\] We need to choose one vowel...
Given: The word \[MONDAY\] Total letters \[=\text{ }6\] (i) \[4\]letters are used at a time Number of ways = (No. of ways of choosing 4 letters from MONDAY) \[=\text{ }{{(}^{6}}{{C}_{4}})\] By using...
Given: Total persons \[=\text{ }10\] Number of persons to be selected \[=\text{ }5\text{ }from\text{ }10\]persons \[({{P}_{1}},\text{ }{{P}_{2}},\text{ }{{P}_{3}}~\ldots \text{ }{{P}_{10}})\] It is...
Given: Total number of vowels \[=\text{ }5\] Total number of consonants \[=\text{ }17\] Number of ways = (No. of ways of choosing 2 vowels from 5 vowels) × (No. of ways of choosing 3 consonants from...
We know that \[3\] points are required to draw a triangle and the collinear points will lie on the same line. Number of triangles formed = (total no. of triangles formed by all 12 points) – (no. of...
(i) a hexagon We know that a hexagon has 6 angular points. By joining those any two angular points we get a line which is either a side or a diagonal. So number of lines formed...
Given: Total number of points \[=\text{ }10\] Number of collinear points \[=\text{ }4\] Number of lines formed = (total no. of lines formed by all 10 points) – (no. of lines formed by collinear...
Given: Total number of questions \[=\text{ }12\] Total number of questions to be answered \[=\text{ }7\] Number of ways = (No. of ways of answering 5 questions from group 1 and 2 from group 2) +...
Given: Total number of questions \[=\text{ }5\] Total number of questions to be answered \[=\text{ }4\] Number of ways = we need to answer 2 questions out of the remaining 3 questions as 1 and 2 are...
Given: Total number of questions \[=\text{ }10\] Questions in part \[A\text{ }=\text{ }6\] Questions in part \[B\text{ }=\text{ }7\] Number of ways = (No. of ways of answering 4 questions from part...
Given: Total number of students in XI \[=\text{ }20\] And, Total number of students in XII \[=\text{ }20\] Total number of students to be selected in a team = 11 (with atleast 5 from class XI and 5...
Given: Total number of officers \[=\text{ }4\] Total number of jawans \[=\text{ }8\] Total number of selection to be made is \[6\] (i) to include exactly one officer Number of ways = (no. of ways of...
Given: Total number of books \[=\text{ }10\] Total books to be selected \[=\text{ }4\] Two particular books are never selected Number of ways = select 4 books out of remaining 8 books as 2 books are...
Given: Total number of books \[=\text{ }10\] Total books to be selected \[=\text{ }4\] (i) there is no restriction Number of ways = choosing 4 books out of 10 books \[={{~}^{10}}{{C}_{4}}\] By using...
Given: Total number of boys \[=\text{ }12\] Total number of girls \[=\text{ }10\] Total number of girls for the competition \[=\text{ }10\text{ }+\text{ }2\text{ }=\text{ }12\] Number of ways = (no....
Given that we need to find the no. of ways of obtaining a product by multiplying two or more from the numbers \[3,\text{ }5,\text{ }7,\text{ }11\] Number of ways = (no. of ways of multiplying two...
As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) ×...
As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) ×...
Given: Total number of players \[=\text{ }16\] Number of players to be selected \[=\text{ }11\] So, the combination is \[^{16}{{C}_{11}}\] By using the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left(...
Given: Total number of courses is \[9\] So out of \[9\text{ }courses\text{ }2\text{ }courses\] are compulsory. Student can choose from \[7\left( i.e.,\text{ }5+2 \right)\] courses only. That too out...
Given: Total boys are \[=\text{ }25\] Total girls are \[=\text{ }10\] Boat party of \[8\]to be made from \[25\] boys and \[10\]girls, by selecting \[5\text{ }boys\text{ }and\text{ }3\text{ }girls\]...
Given: Number of players \[=\text{ }15\] Number of players to be selected \[=\text{ }11\] By using the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\]...
Given: \[^{n\text{ }+\text{ }2}{{C}_{8}}:{{~}^{n\text{ }-\text{ }2}}{{P}_{4}}~=\text{ }57:\text{ }16\] \[^{n\text{ }+\text{ }2}{{C}_{8}}~/{{~}^{n\text{ }-\text{ }2}}{{P}_{4}}~=\text{ }57\text{...
Given: \[^{15}{{C}_{r}}:{{~}^{15}}{{C}_{r\text{ }-\text{ }1}}~=\text{ }11:\text{ }5\] \[^{15}{{C}_{r}}~/{{~}^{15}}{{C}_{r\text{ }-\text{ }1}}~=\text{ }11\text{ }/\text{ }5\] Let us use the formula,...
To find \[r\] let us consider the given expression, \[^{8}{{C}_{r}}-{{~}^{7}}{{C}_{3}}~={{~}^{7}}{{C}_{2}}\] \[^{8}{{C}_{r}}~={{~}^{7}}{{C}_{2}}~+{{~}^{7}}{{C}_{3}}\] We know...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
We know that if \[^{n}{{C}_{p}}~={{~}^{n}}{{C}_{q}}\] then one of the following conditions need to be satisfied: \[\left( i \right)\text{ }p\text{ }=\text{ }q\] \[\left( ii \right)\text{ }n\text{...
Solution: As per the given question
(i) \[^{35}{{C}_{35}}\] Let us use the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\] So now, value of \[n\text{ }=\text{ }35\text{ }and\text{ }r\text{ }=\text{ }35\]...
(i) \[^{14}{{C}_{3}}\] Let us use the formula, \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\] So now, value of \[n\text{ }=\text{ }14\text{ }and\text{ }r\text{ }=\text{ }3\]...