If combinations of letters are formed by taking only $5$ letters at a time out of the letters of the word “METAPHYSICS”. In how many of them will letter T occur.
Answer
363.3k+ views
Hint: Here we want to find a number of ways in which we can make combinations of letters, where T will also occur. So select two alike letters between (S,S), two different letters, then one S and three different letters and last one as no S and 4 different letters, and add these three. You will get the answer.
So we are given the word “METAPHYSICS”.
So we want to find what is given above.
So for that, first, we should find out how many ways the word can be arranged by taking only $5$ letters at a time.
So the formula for permutations with repeated elements is as follows when $k$ out of $n$ elements are indistinguishable. So for example, if we have a total number of books as $n$, with $k$ copies of the same book, the number of different permutations for arranging all the $n$ books is $\dfrac{n!}{k!}$.
Sometimes, we want to count all of the possible ways that a single set of objects can be selected without regard to the order in which they are selected.
A combination is a selection of all or part of a set of objects, without regard to the order in which they were selected. This means that $xyz$ it is considered the same combination $zyx$.
The number of combinations of $n$ objects taken $r$ at a time is denoted by ${}^{n}{{C}_{r}}$.
Where ${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$
So now there are $11$ letters in the word “METAPHYSICS”.
There are $11$ letters T, S, S, and $8$ more letters, all different from each other. Since T is a must, we have to select only $4$ out of the remaining letters that are S, S, and $8$ other different letters.
So in question, it is mentioned that we have to find only combinations.
So no need to consider arrangements.
So now selecting two alike letters from (S, S) and two other different letters from the rest, the number of combinations $={}^{2}{{C}_{2}}{}^{8}{{C}_{2}}$…………. (1)
Next, selecting one S and three other different letters from the rest, the number of combinations $={}^{2}{{C}_{1}}{}^{8}{{C}_{3}}$ ……………(2)
And now selecting no S and four other different letters from the rest, the number of combinations $={}^{8}{{C}_{4}}$ ………………(3)
Hence now total number of combinations, in which T occurs is$={}^{2}{{C}_{2}}{}^{8}{{C}_{2}}+{}^{2}{{C}_{1}}{}^{8}{{C}_{3}}+{}^{8}{{C}_{4}}$
So simplifying further, we get,
$=1\times 28+2\times 56+70$
$=210$ ways.
If combinations of letters are formed by taking only $5$ letters at a time out of all the letters of the word “METAPHYSICS”, then T will occur in $210$ ways.
Note: Read the question in a careful manner. You should know the difference between combinations and arrangements. Also here no arrangements are used only combinations are used. Don’t jumble yourself and confuse between the letters. See what is asked and solve it accordingly. Sometimes silly mistakes occur in this ${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$ avoid the mistakes.
So we are given the word “METAPHYSICS”.
So we want to find what is given above.
So for that, first, we should find out how many ways the word can be arranged by taking only $5$ letters at a time.
So the formula for permutations with repeated elements is as follows when $k$ out of $n$ elements are indistinguishable. So for example, if we have a total number of books as $n$, with $k$ copies of the same book, the number of different permutations for arranging all the $n$ books is $\dfrac{n!}{k!}$.
Sometimes, we want to count all of the possible ways that a single set of objects can be selected without regard to the order in which they are selected.
A combination is a selection of all or part of a set of objects, without regard to the order in which they were selected. This means that $xyz$ it is considered the same combination $zyx$.
The number of combinations of $n$ objects taken $r$ at a time is denoted by ${}^{n}{{C}_{r}}$.
Where ${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$
So now there are $11$ letters in the word “METAPHYSICS”.
There are $11$ letters T, S, S, and $8$ more letters, all different from each other. Since T is a must, we have to select only $4$ out of the remaining letters that are S, S, and $8$ other different letters.
So in question, it is mentioned that we have to find only combinations.
So no need to consider arrangements.
So now selecting two alike letters from (S, S) and two other different letters from the rest, the number of combinations $={}^{2}{{C}_{2}}{}^{8}{{C}_{2}}$…………. (1)
Next, selecting one S and three other different letters from the rest, the number of combinations $={}^{2}{{C}_{1}}{}^{8}{{C}_{3}}$ ……………(2)
And now selecting no S and four other different letters from the rest, the number of combinations $={}^{8}{{C}_{4}}$ ………………(3)
Hence now total number of combinations, in which T occurs is$={}^{2}{{C}_{2}}{}^{8}{{C}_{2}}+{}^{2}{{C}_{1}}{}^{8}{{C}_{3}}+{}^{8}{{C}_{4}}$
So simplifying further, we get,
$=1\times 28+2\times 56+70$
$=210$ ways.
If combinations of letters are formed by taking only $5$ letters at a time out of all the letters of the word “METAPHYSICS”, then T will occur in $210$ ways.
Note: Read the question in a careful manner. You should know the difference between combinations and arrangements. Also here no arrangements are used only combinations are used. Don’t jumble yourself and confuse between the letters. See what is asked and solve it accordingly. Sometimes silly mistakes occur in this ${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$ avoid the mistakes.
Last updated date: 25th Sep 2023
•
Total views: 363.3k
•
Views today: 9.63k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
