The number of ways five alphabets can be chos | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The word $MATHEMATICS$ contains $11$ letters: $M, M, A, A, T, T, H, E, I, C, S$. There are $8$ distinct letters: $\{M, A, T, H, E, I, C, S\}$. We

Vedclass · Accepted Answer

$179$

Vedclass · Answer

(D) The word $MATHEMATICS$ contains $11$ letters: $M, M, A, A, T, T, H, E, I, C, S$. There are $8$ distinct letters: $\{M, A, T, H, E, I, C, S\}$. We need to choose $5$ letters. The cases are: $(1)$ All $5$ letters are distinct: We choose $5$ letters from $8$ distinct letters: $^8C_5 = \frac{8 	imes 7 	imes 6}{3 	imes 2 	imes 1} = 56$. $(2)$ $2$ letters are same (one pair) and $3$ are distinct: There are $3$ pairs $(M, M)$,$(A, A)$,$(T, T)$. We choose $1$ pair in $^3C_1$ ways. From the remaining $7$ distinct letters,we choose $3$ letters in $^7C_3$ ways. Number of ways = $^3C_1 	imes ^7C_3 = 3 	imes 35 = 105$. $(3)$ $2$ pairs of same letters and $1$ distinct letter: We choose $2$ pairs from $3$ available pairs in $^3C_2$ ways. From the remaining $6$ distinct letters,we choose $1$ letter in $^6C_1$ ways. Number of ways = $^3C_2 	imes ^6C_1 = 3 	imes 6 = 18$. Total number of ways = $56 + 105 + 18 = 179$.

The number of ways five alphabets can be chosen from the alphabets of the word $MATHEMATICS$,where the chosen alphabets are not necessarily distinct,is equal to :

Similar Questions

The total number of $3$-digit numbers,whose sum of digits is $10$,is

The greatest positive integer which divides $(n+16)(n+17)(n+18)(n+19)$,for all positive integers $n$,is

The value of $r$ for which $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ^{20}C_{r-2} ^{20}C_2 + ... + ^{20}C_0 ^{20}C_r$ is maximum is

If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module