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Binomial Theorem — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBinomial Theorem4 Marks
Question
How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?
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Answer

The given word is 'MORADABAD'. Number of M = 1, Number of O = 1 Number of R = 1, Number of A = 3 Number of D = 2, Number of B = 1
  1. Number of arrangement of 4 letters.
Selected from these $={^\text{6}}\text{C}_{\text{4}}\times4!$
$=15\times24$
$=360$
  1. Two ailke and with more than one
So, one pair from these and 2 from letters from rest 5 letters.
Number of ways to arrange therefour
$={^\text{2}}\text{C}_{\text{1}}\times​​{^\text{5}}\text{C}_{\text{2}}\times\frac{4!}{2!}$
$=2\times10\times12$
$=240$
  1. Two ailke and with more than
Number of ways to arrange therefour
$={^\text{2}}\text{C}_{\text{2}}\times​​{^\text{5}}\text{C}_{\text{2}}\times\frac{4!}{2!2!}$
$=6$
  1. There alike and one different number of ways to the therefour
$=1\times{^\text{5}}\text{C}_{\text{1}}$
$=5\times\frac{4!}{3!1!}$
$=20$
Required number of ways = 240 + 360 + 6 + 20
Required number of ways = 626

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