Question

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 1

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.