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Centrosomes (Centrioles) are present only in animal cells.  In plants and fungi they are not present.

Centrosomes are near the nucleus of the cell.  They play important role for the cell division.  They initiate the cell division activity.  Centrosomes consist of two mutually perpendicular centrioles and proteins.

Centrioles are part of centrosomes.  They are cylindrical shaped.  They consist of many (nine) (tube shaped) microtubules.  There are two centrioles in a centrosome.  They determine the position of nucleus of cell and spatial distribution of cell.

Their shapes are different.  Centrioles are more important in cell division activity.  Centrioles are not present in plant cells.  They are part of evolution of animal cells to improve efficiency and to decrease errors in mitosis process.

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Centrioles are discs built from microtubule material, with nine microtubules arranged in a cylinder. Each of the nine microtubules has two connected short microtubules. When the cell is not dividing, two centrioles are paired with each other at 90 degree angles. Along with some other proteins, the two centrioles combine to make the centrosome.
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